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Achieving Fully Proportional Representation: Approximability Results

We study the complexity of (approximate) winner determination under the Monroe and Chamberlin--Courant multiwinner voting rules, which determine the set of representatives by optimizing the total (dis)satisfaction of the voters with their representatives. The total (dis)satisfaction is calculated either as the sum of individual (dis)satisfactions (the utilitarian case) or as the (dis)satisfaction of the worst off voter (the egalitarian case). We provide good approximation algorithms for the satisfaction-based utilitarian versions of the Monroe and Chamberlin--Courant rules, and inapproximability results for the dissatisfaction-based utilitarian versions of them and also for all egalitarian cases. Our algorithms are applicable and particularly appealing when voters submit truncated ballots. We provide experimental evaluation of the algorithms both on real-life preference-aggregation data and on synthetic data. These experiments show that our simple and fast algorithms can in many cases find near-perfect solutions.


Introduction
We study the complexity of (approximate) winner determination under the Monroe [32] and Chamberlin-Courant [10] multiwinner voting rules, which aim at selecting a group of candidates that best represent the voters. Multiwinner elections are important both for human societies (e.g., in indirect democracies for electing committees of representatives like parliaments) and for software multiagent systems (e.g., for recommendation systems [25]), and thus it is important to have good multiwinner rules and good algorithms for them. The Monroe and Chamberlin-Courant rules are particularly appealing because they create an explicit (and, in some sense, optimal) connection between the elected committee members and the voters; each voter knows his or her representative and each committee member knows to whom he or she is accountable. In the context of recommendation systems this means that every selected item is personalized, i.e., recommended to a particular user. Moreover, the Monroe rule ensures the proportionality of the representation. We assume that m candidates participate in the election and that the society consists of n voters, who each rank the candidates, expressing their preferences about who they would like to see as their representative.
When choosing a K-member committee, the Monroe and Chamberlin-Courant rules work as follows. For each voter they assign a single candidate as their representative, respecting the following rules: (a) altogether exactly K candidates are assigned to the voters. For the Monroe rule, each candidate is assigned either to about n K voters or to none; for the Chamberlin-Courant rule there is no such restriction and each committee member might be representing a different number of voters. The committee should take this into account in its operation, i.e., by means of weighted voting.
(b) the candidates are selected and assigned to the voters optimally minimizing the total (societal) dissatisfaction or maximizing the total (societal) satisfaction.
The total (dis)satisfaction is calculated on the basis of individual (dis)satisfactions. We assume that there is a function α : N → N such that α(i) measures how well a voter is represented by the candidate that this voter ranks as i'th best. The function α is the same for each voter. We can view α either as a satisfaction function (then it should be a decreasing one) or as a dissatisfaction function (then it should be an increasing one). For example, it is typical to use the Borda count scoring function whose m-candidate dissatisfaction variant is defined as α m B,inc = i − 1, and whose satisfaction variant is α m B,dec = m − i. In the utilitarian variants of the rules, the assignment should maximize (minimize) the total satisfaction (dissatisfaction) calculated as the sum of the voters' individual satisfactions (dissatisfactions) with their representatives. In the egalitarian variants, the assignment should maximize (minimize) the total satisfaction (dissatisfaction) calculated as the satisfaction (dissatisfaction) of the worst-off voter.
The Monroe and Chamberlin-Courant rules create a useful connection between the voters and their representatives that makes it possible to achieve both candidates' accountability to the voters, and proportional representation of voters' views. Among common voting rules, the Monroe and Chamberlin-Courant rules seem to be unique in having both the accountability and the proportionality properties simultaneously. For example, First Past the Post system (where the voters are partitioned into districts with a separate singlewinner Plurality election in each) can give very disproportionate results (forcing some of the voters to be represented by candidates they dislike). On the other side of the spectrum are the party-list systems, which achieve perfect proportionality. In those systems the voters vote for the parties, based on these votes each party receives some number of seats in the parliament, and then each party distributes the seats among its members (usually following a publicly available list of the party's candidates). This makes the elected candidates feel more accountable to apparatchiks of their parties than to the voters. Somewhere between the First Past the Post system and the party-list systems, we have the single transferable vote rule (STV), but for STV it is difficult to tell which candidate represents which voters.
Unfortunately, the Monroe and Chamberlin-Courant rules have one crucial drawback that makes them impractical. It is NP-hard to tell who the winners are! Specifically, NPhardness of winner determination under the Monroe and Chamberlin-Courant rules was shown by Procaccia et al. [37] and by Lu and Boutilier [25]. Worse yet, the hardness holds even if various natural parameters of the election are small [7]. Rare easy cases include those, where the committee to be elected is small, or we consider the Chamberlin-Courant rule and the voters have single-peaked [7] or single-crossing preferences [43].
Lu and Boutilier [25] proposed to use approximation algorithms and have given the first such algorithm for the Chamberlin-Courant system. Their procedure outputs an assignment that achieves no less than 1 − 1 e ≈ 0.63 fraction of the optimal voter satisfaction. However, the approximation ratio 0.63 here means that it is possible that, on average, each agent is represented by a candidate that this agent prefers to only about 63% of the candidates, even if there is a perfect solution that assigns each agent to their most preferred candidate. Such issues, however, would not occurr if we had a constant-factor approximation algorithm minimizing the total dissatisfaction. Indeed, if a perfect solution exists, then the optimal dissatisfaction is zero and a constant-factor approximation algorithm must also output this perfect solution.
The use of approximation algorithms in real-life applications requires some discussion. For example, their use is naturally justified in the context of recommendation systems. Here the strive for optimality is not crucial since a good but not optimal recommendation still has useful information and nobody would object if we replaced the exact recommendation with an approximate one (given that the exact one is hard to calculate). For example, Amazon.com may recommend you a book on gardening which may not be the best book for you on this topic, but still full of useful advice. For such situations, Herbert Simon [41] used the term 'satisficing,' instead of optimizing, to explain the behavior of decision makers under circumstances in which an optimal solution cannot be easily determined. On page 129 he wrote: "Evidently, organisms adapt well enough to satisfice; they do not, in general, 'optimize'." Effectively, what Simon says is that the use of approximation algorithms fits well with the human nature.
Still, the use of approximation algorithms in elections requires some care. It is conceivable that the electoral commission finds an allocation of voters to candidates with a certain value of (dis)satisfaction and one of the parties participating in the election finds an allocation with a better value. This can lead to a political deadlock. There are two ways of avoiding this. Firstly, an approximation algorithm can be fixed by law. In such a case, it becomes an acting voting rule and a new way to measure fairness in the society. Secondly, an electoral commission may calculate the allocation, but also publish the raw data and issue a call for submissions. If, within the period specified by law, nobody can produce a better allocation, then the committee goes ahead and announces the result. If someone produces a better allocation, then the electoral commission uses the latter one.
The use of approximation algorithms is even more natural in elections with partial ballots. Indeed, even if we use an exact algorithm to calculate the winners, the results will be approximate anyway since the voters provide us with approximations of their real preferences and not with their exact preferences.

Our Results
In this paper we focus on approximation algorithms for winner determination under the Monroe and Chamberlin-Courant rules. Our first goal is to seek algorithms that find assignments for which the dissatisfaction of voters is within a fixed bound of the optimal one. Unfortunately, we have shown that under standard complexity-theoretic assumptions such algorithms do not exist. Nonetheless, we found good algorithms that maximize voter's satisfaction. Specifically, we have obtained the following results: 1. The Monroe and Chamberlin-Courant rules are hard to approximate up to any constant factor for the dissatisfaction-based cases (both utilitarian and egalitarian ones; see Theorems 1, 2, 3 and 4) and for the satisfaction-based egalitarian cases (see Theorems 5 and 7). 2. For the satisfaction-based utilitarian framework we show the following. For the Monroe rule with the Borda scoring function we give a (0.715−ǫ)-approximation algorithm (often, the ratio is much better; see Section 4). In case of an arbitrary positional scoring function we give a (1 − 1 e )-approximation algorithm (Theorem 13). For the Chamberlin-Courant rule with the Borda scoring function we give a polynomial-time approximation scheme (that is, for each ǫ, 0 < ǫ < 1, we have a polynomial-time (1 − ǫ)-approximation algorithm; see Theorem 15). 3. We provide empirical evaluation of our algorithms for the satisfaction-based utilitarian framework, both on synthetic and real-life data. This evaluation shows that in practice our best algorithms achieve at least 0.9 approximation ratios, and even better results are typical (see Section 5). 4. We show that our algorithms work very well in the setting where voters do not necessarily rank all the candidates, but only provide the so-called truncated ballots, in which they rank several most preferred candidates (usually at least three). We provide theoretical guarantees on the performance of our algorithms (Propositions 10 and 16) as well as empirical evaluation (see Section 5.4).
Our results show that, as long as one is willing to accept approximate solutions, it is possible to use the utilitarian variants of the Monroe and Chamberlin-Courant rules in practice. This view is justified both from the theoretical and from the empirical point of view. Due to our negative results, we did not perform empirical evaluation for the egalitarian variants of the rules, but we believe that this is an interesting future research direction.

Related Work
A large number of papers are related to our research in terms of methodology (the study of computational complexity and approximation algorithms for winner determination under various NP-hard election rules), in terms of perspective and motivation (e.g., due to the resource allocation view of Monroe and Chamberlin-Courant rules that we take), and in terms of formal similarity (e.g., winner determination under the Chamberlin-Courant rule can be seen as a form of the facility location problem). Below we review this related literature.
There are several single-winner voting rules for which winner determination is known to be NP-hard. These rules include, for example, Dodgson's rule [3,6,20], Young's rule [6,38], and Kemeny's rule [3,5,21]. For the single-transferable vote rule (STV), the winner determination problem becomes NP-hard if we use the so-called parallel-universes tie-breaking [12]. Many of these hardness results hold even in the sense of parameterized complexity theory (however, there also is a number of fixed-parameter tractability results; see the references above for details).
These hardness results motivated the search for approximation algorithms. There are now very good approximation algorithms for Kemeny's rule [1,13,24] and for Dodgson's rule [8,9,16,22,30]. In both cases the results are, in essence, optimal. For Kemeny's rule there is a polynomial-time approximation scheme [24] and for Dodgson's rule the achieved approximation ratio is optimal under standard complexity-theoretic assumptions [8] (unfortunately, the approximation ratio is not constant but depends logarithmically on the number of candidates). On the other hand, for Young's rule it is known that no good approximation algorithms exist [8].
The work of Caragiannis et al. [9] and of Faliszewski et al. [16] on approximate winner determination for Dodgson's rule is particularly interesting from our perspective. In the former, the authors advocate treating approximation algorithms for Dodgson's rule as voting rules in their own right and design them to have desirable properties. In the latter, the authors show that a well-established voting rule (so-called Maximin rule) is a reasonable (though not optimal) approximation of Dodgson's rule. This perspective is important for anyone interested in using approximation algorithms for winner determination in elections (as might be the case for our algorithms for the Monroe and Chamberlin-Courant rules).
The hardness of the winner determination problem for the Monroe and Chamberlin-Courant rules have been considered in several papers. Procaccia, Rosenschein and Zohar [37] were the first to show the hardness of these two rules for the case of a particular approvalstyle dissatisfaction function. Their results were complemented by Lu and Boutilier [25], Betzler, Slinko and Uhlmann [7], Yu, Chan, and Elkind [45], Skowron et al. [43], and Skowron and Faliszewski [42]. These are showing the hardness in case of the Borda dissatisfaction function, obtain results on parameterized hardness of the two rules, and results on hardness (or easiness) for the cases where the profiles are single-peaked or single-crossing. Further, Lu and Boutilier [25] initiated the study of approximability for the Chamberlin-Courant rule (and were the first to use satisfaction-based framework). Specifically, they gave the (1 − 1 e )-approximation algorithm for the Chamberlin-Courant rule. The motivation of Lu and Boutilier was coming from the point of view of recommendation systems and, in that sense, our view of the rules is quite similar to theirs.
In this paper we take the view that the Monroe and Chamberlin-Courant rules are special cases of the following resource allocation problem. The alternatives are shareable resources, each with a certain capacity defined as the maximal number of agents that may share this resource. Each agent has preferences over the resources and is interested in getting exactly one. The goal is to select a predetermined number K of resources and to find an optimal allocation of these resources (see Section 2 for details). This provides a unified framework for the two rules and reveals the connection of proportional representation problem to other resource allocation problems. In particular, it closely resembles multi-unit resource allocation with single-unit demand [40,Chapter 11] (see also the work of Chevaleyre et al. [11] for a survey of the most fundamental issues in the multiagent resource allocation theory) and resource allocation with sharable indivisible goods [2,11]. Below, we point out other connections of the Monroe and Chamberlin-Courant rules to several other problems.
Facility Location Problems. In the facility location problem, there are n customers located in some area and an authority, say a city council, that wants to establish a fixed number k of facilities to serve those customers. Customers incur certain costs (say transportation costs) of using the facilities. Further, setting up a facility costs as well (and this cost may depend on the facility's location). The problem is to find k locations for the facilities that would minimize the total (societal) cost. If these facilities have infinite capacities and can serve any number of customers, then each customer would use his/her most preferred (i.e., closest) facility and the problem is similar to finding the Chamberlin-Courant assignment. If the capacities of the facilities are finite and equal, the problem looks like finding an assignment in the Monroe rule. An essential difference between the two problems are the setup costs and the distance metric. The parameterized complexity of the Facility Location Problem was investigated in Fellows and Fornau [17]. The papers of Procaccia et al. [37] and of Betzler et al. [7] contain a brief discussion of the connection between the Facility Location Problem and the winner determination problem under the Chamberlin-Courant rule.
Group Activity Selection Problem. In the group activity selection problem [14] we have a group of agents (say, conference attendees) and a set of activities (say, options that they have for a free afternoon such as a bus city tour or wine tasting). The agents express preferences regarding the activities and organisers try to allocate agents to activities to maximise their total satisfaction. If there are m possible activities but only k must be chosen by organisers, then we are in the Chamberline-Courant framework, if all activities can take all agents, and in the Monroe framework, if all activities have the same capacities. The difference is that those capacities may be different and also that in the Group Activity Selection Problem we may allow expression of more complicated preferences. For example, an agent may express the following preference "I like wine-tasting best provided that at most 10 people participate in it, and otherwise I prefer a bus city tour provided that at least 15 people participate, and otherwise I prefer to not take part in any activity". The Group Activity Selection Problem is more general than the winner determination in the Monroe and Chamberline-Courant rules. Some hardness and easiness results for this problem were obtained in [14], but the investigation of this problem has only started.
The above connections show that, indeed, the complexity of winner determination under the Monroe and Chamberlin-Courant voting rules are interesting, can lead to progress in several other directions, and may have impact on other applications of artificial intelligence.

Preliminaries
We first define basic notions such as preference orders and positional scoring rules. Then we present our Resource Allocation Problem in full generality and discuss which restrictions of it correspond to the winner determination problem for the Monroe and Chamberlin-Courant voting rules. Finally, we briefly recall relevant notions regarding computational complexity.
Preferences. For each n ∈ N, by [n] we mean {1, . . . , n}. We assume that there is a set N = [n] of agents and a set A = {a 1 , . . . a m } of alternatives. Each alternative a ∈ A has the capacity cap a ∈ N, which gives the total number of agents that can be assigned to it. Further, each agent i has a preference order ≻ i over A, i.e., a strict linear order of the form a π(1) ≻ i a π(2) ≻ i · · · ≻ i a π(m) for some permutation π of [m]. For an alternative a, by pos i (a) we mean the position of a in the i'th agent's preference order. For example, if a is the most preferred alternative for i then pos i (a) = 1, and if a is the least preferred one then pos i (a) = m. A collection V = (≻ 1 , . . . , ≻ n ) of agents' preference orders is called a preference profile.
We will often include subsets of the alternatives in the descriptions of preference orders. For example, if A is the set of alternatives and B is some nonempty strict subset of A, then by B ≻ A − B we mean that for the preference order ≻ all alternatives in B are preferred to those outside of B.
A positional scoring function (PSF) is a function α m : [m] → N. A PSF α m is an increasing positional scoring function (IPSF) if for each i, j ∈ [m], if i < j then α m (i) < α m (j). Analogously, a PSF α m is a decreasing positional scoring function (DPSF) if for Intuitively, if β m is an IPSF then β m (i) can represent the dissatisfaction that an agent suffers when assigned to an alternative that is ranked i'th in his or her preference order. Thus, we assume that for each IPSF β m it holds that β m (1) = 0 (an agent is not dissatisfied by her top alternative). Similarly, a DPSF γ m measures an agent's satisfaction and we assume that for each DPSF γ m it holds that γ m (m) = 0 (an agent is completely not satisfied being assigned his or her least desired alternative). Sometimes we write α instead of α m , when it cannot lead to a confusion.
We will often speak of families α of IPSFs (DPSFs) of the form α = (α m ) m∈N , where α m is a PSF on [m], such that: 2. For a family of DPSFs it holds that α m+1 (i + 1) = α m (i) for all m ∈ N and i ∈ [m].
In other words, we build our families of IPSFs (DPSFs) by appending (prepending) values to functions with smaller domains. To simplify notation, we will refer to such families of IPSFs (DPSFs) as normal IPSFs (normal DPSFs). We assume that each function α m from a family can be computed in polynomial time with respect to m. Indeed, we are particularly interested in the Borda families of IPSFs and DPSFs defined by α m B,inc (i) = i − 1 and α m B,dec (i) = m − i, respectively. Assignment functions. A K-assignment function is any function Φ : N → A, such that Φ(N ) ≤ K (that is, it matches agents to at most K alternatives), and such that for every alternative a ∈ A we have that Φ −1 (a) ≤ cap a (i.e., the number of agents assigned to a does not exceed a's capacity cap a ).
We will also consider partial assignment functions. A partial K-assignment function is defined in the same way as a regular one, except that it may assign a null alternative, ⊥, to some of the agents. It is convenient to think that for each agent i we have pos i (⊥) = m. In general, it might be the case that a partial K-assignment function cannot be extended to a regular one. This may happen, for example, if the partial assignment function uses K alternatives whose capacities sum to less than the total number of voters. However, in the context of Chamberlin-Courant and Monroe rules it is always possible to extend a partial K-assignment function to a regular one.
Given a normal IPSF (DPSF) α, we may consider the following three functions, each assigning a positive integer to any assignment Φ: These functions are built from individual dissatisfaction (satisfaction) functions, so that they can measure the quality of the assignment for the whole society. In the utilitarian framework the first one can be viewed as a total (societal) dissatisfaction function in the IPSF case and a total (societal) satisfaction function in the DPSF case. The second and the third can be used, respectively, as a total dissatisfaction and satisfaction functions for IPSF and DPSF cases in the egalitarian framework. We will omit the word total if no confusion may arise.
For each subset of the alternatives S ⊆ A such that S ≤ K, we denote as Φ S α the partial K-assignment that assigns agents only to the alternatives from S and such that Φ S α maximizes the utilitarian satisfaction ℓ α 1 (Φ S α ). (We introduce this notation only for the utilitarian satisfaction-based setting because it is useful to express appropriate algorithms for this case; for other settings we have hardness results only and this notation would not be useful.) The Resource Allocation Problem. Let us now define the resource allocation problem that forms the base of our study. This problem stipulates finding an optimal K-assignment function, where the optimality is relative to one of the total dissatisfaction or satisfaction functions that we have just introduced. The former is to be minimized and the latter is to be maximized. Definition 1. Let α be a normal IPSF. An instance of α-DU-Assignment problem (i.e., of the disatisfaction-based utilitarian assignment problem) consists of a set of agents N = [n], a set of alternatives A = {a 1 , . . . a m }, a preference profile V of the agents, and a sequence (cap a 1 , . . . , cap am ) of alternatives' capacities. We ask for an assignment function Φ such that: (1) Problem α-SU-Assignment (the satisfaction-based utilitarian assignment problem) is defined identically except that α is a normal DPSF and condition (3) is replaced with "(3 ′ ) ℓ α 1 (Φ) is maximal." If we replace ℓ α 1 with ℓ α ∞ in α-DU-Assignment then we obtain problem α-DE-Assignment, i.e., the dissatisfaction-based egalitarian variant. If we replace ℓ α 1 with ℓ α min in α-SU-Assignment then we obtain problem α-SE-Assignment, i.e., the satisfaction-based egalitarian variant.
Our four problems can be viewed as generalizations of the winner determination problem for the Monroe [32] and Chamberlin-Courant [10] multiwinner voting systems (see the introduction for their definitions). To model the Monroe system, it suffices to set the capacity of each alternative to be N K (for simplicity, throughout the paper we assume that K divides N 1 ). We will refer to thus restricted variants of our problems as the Monroe variants. To represent the Chamberlin-Courant system, we set alternatives' capacities to N . We will refer to the so-restricted variants of our problems as CC variants.
Computational Issues. For many normal IPSFs α and, in particular, for the Borda IPSF, even the above-mentioned restricted versions of the Resource Allocation Problem, namely, α-DU-Monroe, α-DE-Monroe, α-DU-CC, and α-DE-CC are NP-complete [7,37] (the same holds for the satisfaction-based variants of the problems). Thus we seek approximate solutions.
Definition 2. Let r be a real number such that r ≥ 1 (0 < r ≤ 1) and let α be a normal IPSF (a normal DPSF). An algorithm is an r-approximation algorithm for α-DU-Assignment problem (for α-SU-Assignment problem) if on each instance I it returns a feasible assign- where OPT is the optimal total dissatisfaction (satisfaction) ℓ α 1 (Φ OPT ). We define r-approximation algorithms for the egalitarian variants analogously. Lu and Boutilier [25] gave a (1 − 1 e )-approximation algorithm for the SU-CC family of problems. Throughout this paper, we will consider each of the Monroe and CC variants of the problem and for each we will either prove inapproximability with respect to any constant r (under standard complexity-theoretic assumptions) or we will present an approximation algorithm. In our inapproximability proofs, we will use the following two classic NP-complete problems [19]. Definition 4. X3C is a variant of Set-Cover where U is divisible by 3, each member of F has exactly three elements, and K = U 3 . Set-Cover remains NP-complete even if we restrict each member of U to be contained in at most two sets from F (it suffices to note that this restriction is satisfied by Vertex-Cover, which is a special case of Set-Cover). X3C remains NP-complete even if we additionally assume that n is divisible by 2 and each member of U appears in at most 3 sets from F [19].
We will also use results from the theory of parameterized complexity developed by Downey and Fellows [15]. This theory allows to single out a particular parameter of the problem, say k, and analyze its 'contribution' to the overall complexity of the problem. An analogue of the class P here is the class FPT which is the class of problems that can be solved in time f (k)n O(1) , where n is the size of the input instance, and f is some computable function (for a fixed k everything gets polynomial). Parameterized complexity theory also operates with classes W[1] ⊆ W[2] ⊆ · · · which are believed to form a hierarchy of classes of hard problems (combined, they are analogous to the class NP). It holds that FPT ⊆ W[1], but it seems unlikely that FPT = W [1], let alone FPT = W [2]. We point the reader to the books of Niedermeier [34] and Flum and Grohe [18] for detailed overviews of parametrized complexity theory. Interestingly, while both Set-Cover and Vertex-Cover are NPcomplete, the former is W[2]-complete and the latter belongs to FPT (see, e.g., the book of Niedermeier [34] for these now-standard results and their history).

Hardness of Approximation
We now present our inapproximability results for the Monroe and Chamberlin-Courant rules. Specifically, we show that there are no constant-factor approximation algorithms for the dissatisfaction-based variants of the rules (both utilitarian and egalitarian) and for the satisfaction-based egalitarian ones.
Naturally, these inapproximability results carry over to more general settings. For example, unless P = NP, there are no polynomial-time constant-factor approximation algorithms for the general dissatisfaction-based Resource Allocation Problem. On the other hand, our results do not preclude good satisfaction-based approximation algorithms for the utilitarian case and, indeed, in Section 4 we provide such algorithms.  The alternatives from A 2 (positions of one such an alternative is illustrated with the circle) are placed only between the peripheral wavy lines. Each alternative from A 2 is placed on the left from the middle wavy line exactly 2 times, thus each such alternative is placed on the left from the right dashed line no more than 2 times (exactly two times at the figure).
Proof. Let us fix a normal IPSF α and let us assume, aiming at getting a contradiction, that there is some constant r > 1 and a polynomial-time r-approximation algorithm A for α-DU-Monroe.
Let I be an instance of X3C with ground set U = [n] and family F = {F 1 , F 2 , . . . , F m } of 3-element subsets of U . Without loss of generality, we assume that n is divisible by both 2 and 3 and that each member of U appears in at most 3 sets from F.
Using I, we build instance I M of α-DU-Monroe as follows. We set N = U (that is, the elements of the ground set are the agents) and we set A = A 1 ∪A 2 , where A 1 = {a 1 , . . . , a m } is a set of alternatives corresponding to the sets from the family F and A 2 is a set of dummy alternatives of cardinality A 2 = 1 2 n 2 r · α(3), needed for the construction. We let m ′ = A 2 and rename the alternatives in A 2 so that A 2 = {b 1 , . . . , b m ′ }. We set K = n 3 . We build agents' preference orders using the following algorithm.
As the frequency of the elements from U is bounded by 3, we have m f (j) ≤ 3. For each agent j we set his or her preference order to be of the form M f (j) ≻ j A 2 ≻ j M l (j), where the alternatives in M f (j) and M l (j) are ranked in an arbitrary way and the alternatives from A 2 are placed at positions m f (j) + 1, . . . , m f (j) + m ′ in the way described below (see Figure 1 for a high-level illustration of the construction).
We place the alternatives from A 2 in the preference orders of the agents in such a way that for each alternative b i ∈ A 2 there are at most two agents that rank b i among their nr·α(3) top alternatives. The following construction achieves this effect. If (i+j) mod n < 2, then alternative b i is placed at one of the positions m f (j) + 1, . . . , m f (j) + nr · α(3) in j's preference order. Otherwise, b i is placed at a position with index higher than m f (j)+nr·α (3) (and, thus, at a position higher than nr · α (3)). This construction can be implemented because for each agent j there are exactly m ′ · 2 Let Φ be an assignment computed by A on I M . We will show that ℓ α (⇐) If there exists a solution for I (i.e., an exact cover of U with n 3 sets from F), then we can easily show an assignment in which each agent j is assigned to an alternative from the top m f (j) positions of his or her preference order (namely, one that assigns each agent j to the alternative a i ∈ A 1 that corresponds to the set F i , from the exact cover of U , that contains j). Thus, for the optimal assignment Φ OPT it holds that ℓ α In consequence, A must return an assignment with the total dissatisfaction at most nr · α(3).
(⇒) Let us now consider the opposite direction. We assume that A found an assignment Φ such that ℓ α 1 (Φ) ≤ nr · α(3) and we will show that I is a yes-instance of X3C. Since we require each alternative to be assigned to either 0 or 3 agents, if some alternative b i from A 2 were assigned to some 3 agents, at least one of them would rank b i at a position worse than nr · α(3). This would mean that ℓ α 1 (Φ) ≥ nr · α(3) + 1. Analogously, no agent j can be assigned to an alternative that is placed at one of the m l (j) bottom positions of j's preference order. Thus, only the alternatives in A 1 have agents assigned to them and, further, if agents x, y, z, are assigned to some a i ∈ A 1 , then it holds that F i = {x, y, z} (we will call each set F i for which alternative a i is assigned to some agents x, y, z selected). Since each agent is assigned to exactly one alternative, the selected sets are disjoint. Since the number of selected sets is K = n 3 , it must be the case that the selected sets form an exact cover of U . Thus, I is a yes-instance of X3C.
One may wonder if hardness of approximation for α-DU-Monroe is not an artifact of the strict requirements regarding the number of chosen candidates. It turns out that unless P = NP, there is no r-s-approximation algorithm that finds an assignment with the following properties: (1) the aggregated dissatisfaction ℓ α 1 (Φ) is at most r times higher than the optimal one, (2) the number of alternatives to which agents are assigned is at most sK and (3) each selected alternative (the alternative that has agents assigned), is assigned to no more than s⌈ n K ⌉ and no less than 1 s ⌈ n K ⌉ agents. (The proof is similar to the one used for Theorem 1.) Thus, in our further study we do not consider such relaxations of the problem. Proof. The proof of Theorem 1 applies to this case as well. In fact, it even suffices to take m ′ = A 2 = 1 2 nr · α(3). Results analogous to Theorems 1 and 2 hold for the DU-CC family of problems as well.
Theorem 3. For each normal IPSF α and each constant factor r > 1, there is no polynomial-time r-approximation algorithm for α-DU-CC unless P = NP.
Proof. Let us fix a normal IPSF α. For the sake of contradiction, let us assume that there is some constant r > 1, and a polynomial-time r-approximation algorithm A for α-DU-CC. We will show that it is possible to use A to solve the NP-complete Vertex-Cover problem. Let is the ground set, F = {F 1 , . . . , F m } is a family of subsets of U (where each member of U belongs to exactly two sets in F), and K is a positive integer.
Given I, we construct an instance I CC of α-DU-CC as follows. The set of agents is N = U and the set of alternatives is A = m j=1 A j , where each A j contains exactly α(2) · r · n (unique) alternatives. Intuitively, for each j ∈ [m], the alternatives in A j correspond to the set F j . For each A j , 1 ≤ j ≤ m, we pick one alternative, which we denote a j . For each agent i ∈ N , we set i's preference order as follows: Let F j and F k , j < k, be the two sets that contain i.
is irrelevant for the construction). We ask for an assignment of the agents to at most K alternatives.
Let us consider a solution Φ returned by A on input I CC . We claim that ℓ α (⇐) If I is a yes-instance then, clearly, each agent i can be assigned to one of the top two alternatives in his or her preference order (if there is a size-K cover, then this assignment selects at most K candidates). Thus the total dissatisfaction of an optimal assignment is at most n · α(2). As a result, the solution Φ returned by A has total dissatisfaction at most nr · α(2).
(⇒) If A returns an assignment with total dissatisfaction no greater than nr ·α(2), then, by the construction of agents preference orders, we see that each agent i was assigned to an alternative from a set A j such that i ∈ F j . Since the assignment can use at most K alternatives, this directly implies that there is a size-K cover of U with sets from F. Proof. The proof of Theorem 3 is applicable in this case as well. In fact, it even suffices to take the m groups of alternatives, A 1 , . . . , A m , to contain α(2) · r alternatives each.
The above results show that approximating algorithms for finding the minimal dissatisfaction of agents is difficult. On the other hand, if we focus on agents' total satisfaction then constant-factor approximation exist in many cases (see, e.g., the work of Lu and Boutilier [25] and the next section). Yet, if we focus on the satisfaction of the least satisfied voter, there are no efficient constant-factor approximation algorithms for the Monroe and Chamberlin-Courant systems. (However, note that our result for the Monroe setting is more general than the result for the Chamberlin-Courant setting; the latter is for the Borda DPSF only.) Theorem 5. For each normal DPSF α (where each entry is polynomially bounded in the number of alternatives) and each constant factor r, with 0 < r ≤ 1, there is no rapproximation algorithm for α-SE-Monroe unless P = NP.
Proof. Let us fix a DPSF α = (α m ) m∈N , where each entry α m is polynomially bounded in the number of alternatives m. For the sake of contradiction, let us assume that for some r, 0 < r ≤ 1, there is a polynomial-time r-approximation algorithm A for α-SE-Monroe. We will show that the existence of this algorithm implies that X3C is solvable in polynomial time.
Let I be an X3C instance with ground set U = {1, 2, . . . , n} and collection F = {F 1 , . . . , F m } of subsets of U . Each set in F has cardinality three. Further, without loss of generality, we can assume that n is divisible by three and that each i ∈ U appears in at most three sets from F. Given I, we form an instance I M of α-SE-Monroe as follows.
. The set N of agents is partitioned into two subsets, N 1 and N 2 . N 1 contains n agents (intuitively, corresponding to the elements of the ground set U ) and N 2 contains n ′ agents (used to enforce certain properties of the solution). The set of alternatives A is partitioned into two subsets, A 1 and A 2 . We set A 1 = {a 1 , . . . , a m } (members of A 1 correspond to the sets in F), and we set For each j, 1 ≤ j ≤ n, we set the preference order of the j'th agent in N 1 to be of the form Note that by our assumptions, M f (j) ≤ 3. For each j, 1 ≤ j ≤ n ′ , we set the preference order of the j'th agent in N 2 to be of the form Note that each agent in N 2 ranks the alternatives from A 1 in positions m ′ + 1, . . . , m ′ + m. Finally, we set the number of candidates that can be selected to be K = n+n ′ 3 . Now, consider the solution Φ returned by A on I M . We will show that (⇐) If there exists an exact set cover of U with sets from F, then it is easy to construct a solution for I M where the satisfaction of each agent is greater or equal to r · α m+m ′ (3). Let I ⊆ {1, . . . , m} be a set such that i∈I F i = U and I = n 3 . We assign each agent j from N 1 to the alternative a i such that (a) i ∈ I and (b) j ∈ F i , and we assign each agent from N 2 to his or her most preferred alternative. Thus, Algorithm A has to return an assignment with the minimal satisfaction greater or equal to r · α m+m ′ (3).
(⇒) For the other direction, we first show that r · α m+m ′ (3) ≥ α m+m ′ (m ′ ). Since DPSFs are strictly decreasing, it holds that: Then, by the definition of DPSFs, it holds that: Using the fact that m ′ = (α m+1 (1)·⌈ 1−r r ⌉+3) and using (2), we can transform inequality (1) to obtain the following: This means that if the minimal satisfaction of an agent is at least r · α m+m ′ (3), then no agent was assigned to an alternative that he or she ranked beyond position m ′ . If some agent j from N 1 were assigned to an alternative from A 2 , then, by the pigeonhole principle, some agent from N 2 would be assigned to an alternative from A 1 . However, each agent in N 2 ranks the alternatives from A 1 beyond position m ′ and thus such an assignment is impossible. In consequence, it must be that each agent in j was assigned to an alternative that corresponds to a set F i in F that contains j. Such an assignment directly leads to a solution for I.
Let us now move on to the case of SE-CC family of problems. Unfortunately, in this case our inapproximability argument holds for the case of Borda DPSF only (though we believe that it can be adapted to other DPSFs as well). Further, in our previous theorems we were showing that existence of a respective constant-factor approximation algorithm implies that NP collapses to P. In the following theorem we will show a seemingly weaker collapse of W [2] to FPT.
To prove hardness of approximation for α B,dec -SE-CC, we first prove the following simple lemma. Lemma 6. Let K, p, l be three positive integers and let X be a set of cardinality lpK.

There exists a family
and let Y ′ be a family of all K-element subsets of X ′ . Replace each element i of X ′ with p new elements (at the same time replacing i with the same p elements within each set in Y ′ that contains i). As a result we obtain two new sets, X and Y , that satisfy the statement of the theorem (up to the renaming of the elements).
Proof. For the sake of contradiction, let us assume that there is some constant r, 0 < r ≤ 1, and a polynomial-time r-approximation algorithm A for α m B,dec -SE-CC. We will show that the existence of this algorithm implies that Set-Cover is fixed-parameter tractable for the parameter K (since Set-Cover is known to be W[2]-complete for this parameter, this will imply FPT = W [2]).
Let I be an instance of Set-Cover with ground set U = [n] and family F = {F 1 , F 2 , . . . , F m } of subsets of U . Given I, we build an instance I CC of α m B,dec -SE-CC as follows. The set of agents N consists of n subsets of agents, N 1 , . . . , N n , where each group N i contains exactly n ′ = ⌈ 2 r ⌉K K agents. Intuitively, for each i, 1 ≤ i ≤ n, the agents in the set N i correspond to the element i in U . The set of alternatives A is partitioned into two subsets, A 1 and A 2 , such that: (1) A 1 = {a 1 , . . . , a m } is a set of alternatives corresponding to the sets from the family F, and (2) Before we describe the preference orders of the agents in N , we form a family R = {r 1 , . . . , r n ′ } of preference orders over A 2 that satisfies the following condition: For each K-element subset B of A 2 , there exists r j in R such that all members of B are ranked among the bottom m(1+r) K K positions in r j . By Lemma 6, such a construction is possible (it suffices to take l = 2 r and p = m(1+r) K ); further, the proof of the lemma provides an algorithmic way to construct R.
We form the preference orders of the agents as follows.
For each i, 1 ≤ i ≤ n, and each j, 1 ≤ j ≤ n ′ , the j'th agent from N i has preference order of the form: (we pick any arbitrary, polynomial-time computable order of candidates within M f (i) and M l (i)).
Let Φ be an assignment computed by A on I M . We will show that ℓ if and only if I is a yes-instance of Set-Cover.
(⇐) If there exists a solution for I (i.e., a cover of U with K sets from F), then we can easily show an assignment where each agent is assigned to an alternative that he or she ranks among the top m positions (namely, for each j, 1 ≤ j ≤ n, we assign all the agents from the set N j to the alternative a i ∈ A 1 such that j ∈ F i and F i belongs to the alleged K-element cover of U ). Under this assignment, the least satisfied agent's satisfaction is at least m ′ − m and, thus, A has to return an assignment Φ where ℓ (⇒) Let us now consider the opposite direction. We assume that A found an assignment and we will show that I is a yes-instance of Set-Cover. We claim that for each i, 1 ≤ i ≤ n, at least one agent j in N i were assigned to an alternative from A 1 . If all the agents in N i were assigned to alternatives from A 2 , then, by the construction of R, at least one of them would have been assigned to an alternative that he or she ranks at a position greater than A 2 − m(1+r) (we skip the straightforward calculation) and, thus, this agent would have been assigned to an alternative that he or she ranks at a position greater than m ′ − m ′ r + mr. As a consequence, this agent's satisfaction would be lower than (m ′ − m)r. Similarly, no agent from N i can be assigned to an alternative from M l (i). Thus, for each i, 1 ≤ i ≤ n, there exists at least one agent j ∈ N i that is assigned to an alternative from M f (i). In consequence, the covering subfamily of F consists simply of those sets F k , for which some agent is assigned to alternative a k ∈ A 1 . The presented construction gives the exact algorithm for Set-Cover problem running The existence of such an algorithm means that Set-Cover is in FPT. On the other hand, we know that Set-Cover is W[2]-complete, and thus if A existed then FPT = W [2] would hold.

Algorithms for the Utilitarian, Satisfaction-Based Cases
We now turn to approximation algorithms for the Monroe and Chamberlin-Courant multiwinner voting rules in the satisfaction-based framework. Indeed, if one focuses on agents' total satisfaction then it is possible to obtain high-quality approximation results. In particular, we show the first nontrivial (randomized) approximation algorithm for α B,dec -SU-Monroe. We show that for each ǫ > 0 we can provide a randomized polynomial-time algorithm that achieves 0.715 − ǫ approximation ratio; the algorithm usually gives even better approximation guarantees. For the case of arbitrarily selected DPSF we show a (1 − e −1 )-approximation algorithm. Finally, we show the first polynomial-time approximation scheme (PTAS) for α B,dec -SU-CC. These results stand in sharp contrast to those from the previous section, where we have shown that approximation is hard for essentially all remaining variants of the problem.
The core difficulty in solving α-Monroe/CC-Assignment problems lays in selecting the alternatives that should be assigned to the agents. Given a preference profile and a set S of up to K alternatives, using a standard network-flow argument, it is easy to find a (possibly partial) optimal assignment Φ S α of the agents to the alternatives from S.  N ). Then there is a polynomial-time algorithm that computes a (possibly partial) optimal assignment Φ S α of the agents to the alternatives from S.
Notation: Φ ← a map defining a partial assignment, iteratively built by the algorithm. Φ ← ← the set of agents for which the assignment is already defined. Φ → ← the set of alternatives already used in the assignment. if K ≤ 2 then compute the optimal solution using an algorithm of Betzler et al. [7] and return. Note that for the case of the Chamberlin-Courant rule the algorithm from the above proposition can be greatly simplified: To each voter we assign the candidate that he or she ranks highest among those from S. For the case of Monroe, unfortunately, we need the expensive network-flow-based approach. Nonetheless, Proposition 8 allows us to focus on the issue of selecting the winning alternatives and not on the issue of matching them to the agents.
Below we describe our algorithms for α B,dec -SU-Monroe and for α B,dec -SU-CC. Formally speaking, every approximation algorithm for α B,dec -SU-Monroe also gives feasible results for α B,dec -SU-CC. However, some of our algorithms are particularly well-suited for both problems and some are tailored to only one of them. Thus, for each algorithm we clearly indicate if it is meant only for the case of Monroe, only for the case of CC, or if it naturally works for both systems.

Algorithm A (Monroe)
Perhaps the most natural approach to solve α B,dec -SU-Monroe is to build a solution iteratively: In each step we pick some not-yet-assigned alternative a i (using some criterion) and assign it to those ⌈ N K ⌉ agents that (a) are not assigned to any other alternative yet, and (b) whose satisfaction of being matched with a i is maximal. It turns out that this idea, implemented formally as Algorithm A (see pseudo code in Figure 2), works very well in many cases. We provide a lower bound on the total satisfaction it guarantees in the next lemma. We remind the reader that the so-called k'th harmonic number H k = k i=1 1 i has asymptotics H k = Θ(log k).
Proof. Our algorithm explicitly computes an optimal solution when K ≤ 2 so we assume that K ≥ 3. Let us consider the situation in the algorithm after the i'th iteration of the outer loop (we have i = 0 if no iteration has been executed yet). So far, the algorithm has picked i alternatives and assigned them to i n K agents (recall that for simplicity we assume that K divides n evenly). Hence, each agent has ⌈ m−i K−i ⌉ unassigned alternatives among his or her i + ⌈ m−i K−i ⌉ top-ranked alternatives. By pigeonhole principle, this means that there is an unassigned alternative a ℓ who is ranked among top i + ⌈ m−i K−i ⌉ positions by at least n K agents. To see this, note that there are (n − i n K )⌈ m−i K−i ⌉ slots for unassigned alternatives among the top i + ⌈ m−i K−i ⌉ positions in the preference orders of unassigned agents, and that there are m − i unassigned alternatives. As a result, there must be an alternative a ℓ for whom the number of agents that rank him or her among the top i + ⌈ m−i K−i ⌉ positions is at least: In consequence, the ⌈ n K ⌉ agents assigned in the next step of the algorithm will have the total satisfaction at least ⌈ n Thus, summing over the K iterations, the total satisfaction guaranteed by the assignment Φ computed by Algorithm Ais at least the following value: (to derive the fifth line from the fourth one we note that K(H K −1)−H K ≥ 0 when K ≥ 3): If each agent were assigned to his or her top alternative, the total satisfaction would be equal to (m − 1)n. Thus we get the following bound: This completes the proof.
Note that in the above proof we measure the quality of our assignment against, a perhaps-impossible, perfect solution, where each agent is assigned to his or her top alternative. This means that for relatively large m and K, and small K m ratio, the algorithm can achieve a close-to-ideal solution irrespective of the voters' preference orders. We believe that this is an argument in favor of using Monroe's system in multiwinner elections. On the flip side, to obtain a better approximation ratio, we would have to use a more involved bound on the quality of the optimal solution. To see that this is the case, form an instance I of α B,dec -SU-Monroe with n agents and m alternatives, where all the agents have the same preference order, and where we seek to elect K candidates (and where K divides n). It is easy to see that each solution that assigns the K universally top-ranked alternatives to the agents is optimal. Thus the total dissatisfaction of the agents in the optimal solution is: .
By taking large enough m and K (even for a fixed value of m K ), the fraction 1 − K−1

2(m−1)
can be arbitrarily close to the approximation ratio of our algorithm (the reasoning here is somewhat in the spirit of the idea of identifying maximally robust elections, as studied by Shiryaev, Yu, and Elkind [39]).
For small values of K, it is possible that the H K K part of our approximation ratio would dominate the K−1 2(m−1) part. In such cases we can use the result of Betzler et al. [7], who showed that for each fixed constant K, α B,dec -SU-Monroe can be solved in polynomial time. Thus, for the finite number of cases where H K K is too large, we can solve the problem optimally using their algorithm. In consequence, the quality of the solution produced by Algorithm A most strongly depends on the ratio K−1 2(m−1) . In most cases we can expect it to be small (for example, in Polish parliamentary elections K = 460 and m ≈ 6000; in this case the greedy algorithm's approximation ratio is about 0.96).
Our algorithm has one more great advantage: Since it focuses on the top parts of voters' preference orders, it can achieve very good results even if the voters submit so-called truncated ballots (that is, if they rank some of their top alternatives only). Below we present the formal analysis of the algorithm's approximation ratio for this case. Unfortunately, we did not obtain a closed form formula and, instead, we present the guaranteed approximation ratio as a sum, in Proposition 10 below. We also present the relation between the fraction of the top alternatives ranked by each of the voters and the approximation ratio for few values of m and K in Figure 3. and Proof. We use the same approach as in the proof of Lemma 9, except that we adjust our estimates of voters' satisfaction. Consider a situation after some i'th iteration of the algorithm's outer loop (i = 0 if we are before the first iteration). If i + m−i K−i ≤ P , then we can use the same lower bound for the satisfaction of the agents assigned in the (i + 1)'th iteration as in the proof of Lemma 9. That is, the agents assigned in the (i + 1)'th iteration will have satisfaction at least r 1 For the case where i + m−i K−i > P , the bound from Lemma 9 does not hold, but we can use a similar approach to find a different one. Let P x ≤ P be some positive integer. We are interested in the number x of not-yet assigned agents who rank some not-yet-selected alternative among their top P x positions (after the i'th iteration). Similarly as in the proof of Lemma 9, using the pigeonhole principle we note that: Thus, the satisfaction of the agents assigned in the (i + 1)'th iteration is at least: The implies that i + m−i K−i ≤ P and for this case we lower-bound agents' satisfaction by r 1 (i). For the case where (K−i)(Px−i) We use this estimate for the satisfaction of the agents assigned in the (i + 1)'th iteration for the cases where (a) i ). In this case we estimate (4) as follows: For the remaining cases, we set P x = P and (4) becomes: Naturally, we replace our estimates by 0 whenever they become negative.
To complete the proof, it suffice to, as in the proof of Lemma 9, note that (m − 1)n is an upper bound on the satisfaction achieved by the optimal solution. Our results show that for most settings there is very little reason to ask the agents to rank all the alternatives. Using Proposition 10, election designers can estimate how many alternatives the agents should rank to obtain a particular level of satisfaction and, since computing preference orders can be expensive for the agents, this way can save a large amount of effort.

Algorithm B (Monroe)
There are simple ways in which we can improve the quality of the assignments produced by Algorithm A. For example, our Algorithm B first runs Algorithm A and then, using Proposition 8, optimally reassigns the alternatives to the voters. As shown in Section 5, this very simple trick turns out to noticeably improve the results of the algorithm in practice (and, of course, the theoretical approximation guarantees of Algorithm A carry over to Algorithm B).

Algorithm C (Monroe, CC)
Algorithm C is a further heuristic improvement over Algorithm B. This time the idea is that instead of keeping only one partial function Φ that is iteratively extended up to the full Notation: We use the same notation as in Algorithm A; P ar ← a list of partial representation functions P ar = [] P ar.push({}) ← a i newP ar.push(Φ ′ ) sort newP ar according to descending order of the total satisfaction of the assigned agents P ar ← chose first d elements of newP ar for Φ ∈ P ar do Φ ← compute the optimal representative function using an algorithm of Betzler et al. [7] for the set of winners Φ → return the best representative function from P ar assignment, we keep a list of up to d partial assignment functions, where d is a parameter of the algorithm. At each iteration, for each assignment function Φ among the d stored ones and for each alternative a that does not yet have agents assigned to by this Φ, we compute an optimal extension of this Φ that assigns agents to a. As a result we obtain possibly more than d (partial) assignment functions. For the next iteration we keep those d of them that give highest satisfaction.
We provide pseudocode for Algorithm C in Figure 4. If we take d = 1, we obtain Algorithm B. If we also disregard the last two lines prior to returning the solution, we obtain Algorithm A.
Algorithm C can also be adapted for the Chamberlin-Courant rule. The only difference concerns creating the assignment functions: we replace the contents of the first foreach loop with the following code: Note that, for the case of the Chamberlin-Courant rule, Algorithm C can also be seen as a generalization of Algorithm GM that we will discuss later in Section 4.5.

Algorithm R (Monroe, CC)
Algorithms A, B and C achieve very high approximation ratios for the cases where K is small relative to m. For the remaining cases, where K and m are comparable, we can use a sampling-based randomized algorithm (denoted as Algorithm R) described below. We focus on the case of Monroe and we will briefly mention the case of CC at the end.
The idea of this algorithm is to randomly pick K alternatives and match them optimally to the agents, using Proposition 8. Naturally, such an algorithm might be very unlucky and pick K alternatives that all of the agents rank low. Yet, if K is comparable to m then it is likely that such a random sample would include a large chunk of some optimal solution. In the lemma below, we asses the expected satisfaction obtained with a single sampling step (relative to the satisfaction given by the optimal solution) and the probability that a single sampling step gives satisfaction close to the expected one. Naturally, in practice one should try several sampling steps and pick the one with the highest satisfaction.

Lemma 11. A single sampling step of the randomized algorithm for α B,dec -SU-Monroe achieves expected approximation ratio of
. Let p ǫ denote the probability that the relative deviation between the obtained total satisfaction and the expected total satisfaction is higher than ǫ. Then for K ≥ 8 we have p ǫ ≤ exp − Kǫ 2 128 .
Proof. Let N = [n] be the set of agents, A = {a 1 , . . . , a m } be the set of alternatives, and V be the preference profile of the agents. Let us fix some optimal solution Φ opt and let A opt be the set of alternatives assigned to the agents in this solution. For each a i ∈ A opt , we write sat(a i ) to denote the total satisfaction of the agents assigned to a i in Φ opt . Naturally, we have a∈Aopt sat(a) = OPT. In a single sampling step, we choose uniformly at random a K-element subset B of A. Then, we form a solution Φ B by matching the alternatives in B optimally to the agents (via Proposition 8). We write K opt to denote the random variable equal to A opt ∩ B , the number of sampled alternatives that belong to A opt . We define p i = Pr(K opt = i). For each j ∈ {1, . . . , K}, we write X j to denote the random variable equal to the total satisfaction of the agents assigned to the j'th alternative from the sample. We claim that for each i, 0 ≤ i ≤ K, it holds that: Why is this so? Given a sample B that contains i members of A opt , our algorithm's solution is at least as good as a solution that matches the alternatives from B ∩ A opt in the same way as Φ opt , and the alternatives from B − A opt in a random manner. Since K opt = i and each a j ∈ A opt has equal probability of being in the sample, it is easy to see that the expected value of a j ∈B∩Aopt sat(a j ) is i K OPT. After we allocate the agents from B ∩ A opt , each of the remaining, unassigned agents has m − i positions in his or her preference order where he ranks the agents from A − A opt . For each unassigned agents, the average score value associated with these positions is at least m−i−1 agent could rank the alternatives from B ∩ A opt in the top i positions). There are (n − i n K ) such not yet assigned agents and so the expected total satisfaction from assigning them randomly to the alternatives is m−i−1 2 · (n − i n K ). This proves our bound on the expected satisfaction of a solution yielded by optimally matching a random sample of K alternatives.
Since OPT is upper bounded by (m − 1)n (consider a possibly-nonexistent solution where every agent is assigned to his or her top preference), we get that: We can compute the unconditional expected satisfaction of Φ B as follows: Since K i=1 p i ·i is the expected number of the alternatives in A opt , we have that K i=1 p i ·i = In consequence, through routine calculation, we get that: It remains to assess the probability that the total satisfaction obtained through Φ B is close to its expected value. Since X j ∈ 0, (m−1)n K , from Hoeffding's inequality we get: We note that since K m − K 2 m 2 −m ≥ 0, our previous calculations show that E ( K j=1 X j ) ≥ OPT 2 . Further, for K ≥ 8, Lemma 9 (and the fact that in its proof we upper-bound OPT to be (m − 1)n) gives that OPT ≥ mn 8 . Thus p ǫ ≤ exp − Kǫ 2 128 . This completes the proof.
In the next theorem we will see that to have a high chance of obtaining a high quality assignment, we need to repeat the sampling step many times. Thus, for practical purposes, by Algorithm R we mean an algorithm that repreats the sampling process a given number of times (this parameter is given as input) and returns the best solution found (the assignment is created using Proposition 8).
The threshold for K m , where the sampling step is (in expectation) better than the greedy algorithm is about 0.57. Thus, by combining the two algorithms, we can guarantee an expected approximation ratio of 0.715 − ǫ, for each fixed constant ǫ. The pseudo-code of the combination of the two algorithms (Algorithm AR) is presented in Figure 5. Proof. Let ǫ be a fixed constant. We are given an instance I of α B,dec -SU-Monroe. If m ≤ 1 + 2 ǫ , we solve I using a brute-force algorithm (note that in this case the number of alternatives is at most a fixed constant). Similarly, if H K K ≥ ǫ 2 then we use the exact algorithm of Betzler et al. [7] for a fixed value of K (note that in this case K is no greater than a certain fixed constant). We do the same if K ≤ 8.
On the other hand, if neither of the above conditions hold, we try both Algorithm A and a number of runs of the sampling-based algorithm. It is easy to check through routine calculation that if H K K ≤ ǫ 2 and m > 1 + 2 ǫ then Algorithm A achieves approximation ratio no worse than (1 − K 2m − ǫ). We run the sampling-based algorithm −512 log(1−λ) times. The probability that a single run fails to find a solution with approximation ratio at least . Thus, the probability that at least one run will find a solution with at least this approximation ratio is at least: Since m ≤ 1 + 2 ǫ , by routine calculation we see that the sampling-based algorithm with probability λ finds a solution with approximation ratio at least 1 2 (1 + K m − K 2 m 2 + K 3 m 3 ) − ǫ. By solving the equality: we can find the value of K m for which the two algorithms give the same approximation ratio. By substituting x = K m we get equality 1 + x − x 2 + x 3 = 2 − x. One can calculate that this equality has a single solution within 0, 1 and that this solution is x ≈ 0.57. For this x both algorithms guarantee approximation ratio of 0.715 − ǫ. For x < 0.57 the deterministic algorithm guarantees a better approximation ratio and for x > 0.57, the randomized algorithm does better. Notation: We use the same notation as in Algorithm 2; w(·) denotes Lambert's W-Function. Parameters: λ ← required probability of achieving the approximation ratio equal 0.715 − ǫ if HK K ≥ ǫ 2 or K ≤ 8 then compute the optimal solution using an algorithm of Betzler et al. [7] and return. if m ≤ 1 + 2 ǫ then compute the optimal solution using a simple brute force algorithm and return. Φ 1 ← solution returned by Algorithm A Φ 2 ← run the sampling-based algorithm −512 log(1−λ) Kǫ 2 times; select the assignment of the best quality return the better assignment among Φ 1 and Φ 2 Let us now consider the case of CC. It is just as natural to try a sampling-based approach for solving α B,dec -SU-CC, as we did for the Monroe variant. Indeed, as recently (and independently) observed by Oren [35], this leads to a randomized algorithm with expected approximation ratio of (1 − 1 K+1 )(1 + 1 m ). However, since we will later see an effective, deterministic, polynomial-time approximation scheme for α B,dec -SU-CC, there is little reason to explore the sampling based approach.

Algorithm GM (Monroe, CC)
Algorithm GM (greedy marginal improvement) was introduced by Lu and Boutilier for the case of the Chamberlin-Courant rule. Here we generalize it to apply to Monroe's rule as well, and we show that it is a 1 − 1 e approximation algorithm for α-SU-Monroe. We point out that this approximation result for Monroe rule applies to all non-decreasing PSFs α. For the Monroe rule, the algorithm can be viewed as an extension of Algorithm B.
The algorithm proceeds as follows. We start with an emtpy set S. Then we execute K iterations. In each iteration we find an alternative a that is not assigned to agents yet, and that maximizes the value Φ S∪{a} α . (A certain disadvantage of this algorithm for the case of Monroe is that it requires a large number of computations of Φ S α ; since in Monroe's rule each alternative can be assigned at most n K agents in the partial assignment Φ S α , computation of Φ S α is a slow process based on min-cost/max-flow algorithm.) We provide the pseudocode for Algorithm GM in Figure 6. Proof. The proof follows by applying the powerful result of Nemhauser et al. [33], which says that greedy algorithms achieve 1− 1 e approximation ratio when used to optimize nondecreasing submodular functions (we explain these notions formally below). The main challenge in the proof is to define a function that, on one hand, satisfies the conditions of Nemhauser et al.'s result, and, on the other, models solutions for α-SU-Monroe.
Notation: Φ S α -the partial assignement that assigns a single alternative to at most ⌈ n K ⌉ agents, that assigns to the agents only the alternatives from S, and that maximizes the utilitarian satisfaction ℓ α Figure 6: Pseudocode for Algorithm GM.
Let A be a set of alternatives, N = [n] be a set of agents with preferences over A, α be an A -candidate DPSF, and K ≤ A be the number of representatives that we want to elect. We consider function z : 2 A → N defined, for each set S, S ⊆ A and S ≤ K, as z(S) = ℓ α 1 (Φ S α ). Clearly, z(S) is nondecreasing (that is, for each two sets A and B, if A ⊆ B and B ≤ K then z(A) ≤ z(B). Since argmax S⊂A, S =K z(S) is the set of winners under α-Monroe and since Algorithm GM builds the solution iteratively by greedily extending initially empty set S so that each iteration increases the value of z(S) maximally, if z were submodular then by the results of Nemhauser et al. [33] we would get that Algorithm GM is a (1 − 1 e )-approximation algorithm. Thus, our goal is to show that z is submodular. Formally, our goal is to show that for each two sets S and T , S ⊂ T , and each alternative a / ∈ T it holds that z(S ∪ {a}) − z(S) ≥ z(T ∪ {a}) − z(T ) (this is the formal definition of submodularity). First, we introduce a notion that generalizes the notion of a partial set of winners S. Let s : A → N denote a function that assigns a capacity to each alternative (i.e., s gives a bound on the number of agents that a given alternative can represent). Intuitively, each set S, S ⊆ A, corresponds to the capacity function that assigns ⌈ n k ⌉ to each alternative a ∈ S and 0 to each a / ∈ S. Given a capacity function s, we define a partial solution Φ s α to be one that maximizes the total satisfaction of the agents and that satisfies the new capacity constraints: ∀ a∈S (Φ s α ) −1 (a) ≤ s(a). To simplify notation, we write s ∪ {a} to denote the function such that (s ∪ {a})(a) = s(a) + 1 and ∀ a ′ ∈S\{a} (s ∪ {a})(a ′ ) = s(a ′ ). (Analogously, we interpret s \ {a} as subtracting one from the capacity for a; provided it is nonzero.) Also, by s ≤ t we mean that ∀ a∈A s(a) ≤ t(a). We extend our function z to allow us to consider a subset of the agents only. For each subset N ′ of the agents and each capacity function s, we define z N ′ (s) to be the satisfaction of the agents in N ′ obtained under Φ s α . We will now prove a stronger variant of submodularity for our extended z. That is, we will show that for each two capacity functions s and t it holds that: Our proof is by induction on N . Clearly, Equation (5) holds for N ′ = ∅. Now, assuming that Equation (5) holds for every N ′ ⊂ N we will prove its correctness for N . Let i denote an agent such that Φ Let a s = Φ s α (i) and a t = Φ t α (i). We have: We also have: Since Φ t α describes an optimal representation function under the capacity restrictions t, we have that: Finally, from the inductive hypothesis for N ′ = N \ {i} we have: By combining these inequalities we get: This completes the proof.
Formally speaking, Algorithm GM is never worse than Algorithm A. For Borda satisfaction function, it inherits the approximation guarantees from Algorithm A, and for other cases Theorem 13 guarantees approximation ratio 1 − 1 e (we do not know of any guarantees for Algorithm A for these cases). The comparison with Algorithms B and C is not nearly as easy. Algorithm GM is still likely better than them for satisfaction functions significantly different from Borda's, but for the Borda case our experiments show that Algorithm GM is much slower than Algorithms B and C and obtains almost the same or slightly worse results (see Section 5).

Algorithm P (CC)
The idea of our algorithm (presented in Figure 7) is to compute a certain value x and to greedily compute an assignment that (approximately) maximizes the number of agents assigned to one of their top-x alternatives. 2 If after this process some agent has no alternative assigned, we assign him or her to his or her most preferred alternative from those already picked. Somewhat surprisingly, it turns out that this greedy strategy achieves high-quality results. (Recall that for nonnegative real numbers, Lambert's W-function, w(x), is defined to be the solution of the equation x = w(x)e w(x) .) Notation: We use the same notation as in Algorithm C; num pos x (a) ← {i ∈ [n] \ Φ ← : pos i (a) ≤ x} (the number of not-yet assigned agents that rank alternative a in one of their first x positions) Proof. Let x = mw(K) K . We will first give an inductive proof that, for each i, 0 ≤ i ≤ K, after the i'th iteration of the outer loop at most n(1 − w(K) K ) i agents are unassigned. Based on this observation, we will derive the approximation ratio of our algorithm.
For i = 0, the inductive hypothesis holds because n(1 − w(K) K ) 0 = n. For each i, let n i denote the number of unassigned agents after the i'th iteration. Thus, after the i'th iteration there are n i unassigned agents, each with x unassigned alternatives among his or her top-x ranked alternatives. As a result, at least one unassigned alternative is present in at least n i x m−i of top-x positions of unassigned agents. This means that after the (i + 1)'st iteration the number of unassigned agents is: If for a given i the inductive hypothesis holds, that is, if n i ≤ n 1 − w(K) K i , then: Thus the hypothesis holds and, as a result, we have that: Let Φ be the assignment computed by our algorithm. To compare it against the optimal solution, it suffices to observe that the optimal solution has the value of satisfaction of at most OPT ≤ (m − 1)n, that each agent selected during the first K steps has satisfaction at least m − x = m − mw(K) K , and that the agents not assigned within the first K steps have satisfaction no worse than 0. Thus it holds that: This completes the proof.
Since for each ǫ > 0 there is a value K ǫ such that for each K > K ǫ it holds that 2w(K) K < ǫ, and α B,dec -SU-CC problem can be solved optimally in polynomial time for each fixed constant K(see the work of Betzler et al. [7]), there is a polynomial-time approximation scheme (PTAS) for α B,dec -SU-CC (i.e., a family of algorithms such that for each fixed r, 0 < r < 1, there is a polynomial-time r-approximation algorithm for α B,dec -SU-CC in the family; note that in PTASes we measure the running time by considering r to be a fixed constant).

Theorem 15. There is a PTAS for α B,dec -SU-CC.
The idea used in Algorithm P can also be used to address a generalized SE-CC problem. We can consider the following relaxation of SE-CC: Instead of requiring that each agent's satisfaction is lower-bounded by some value, we ask that the satisfactions of a significant majority of the agents are lower-bounded by a given value. More formally, for a given constant δ, we introduce an additional quality metric: For a given 0 < δ < 1, by putting x = −m ln(δ) K , we get (1 + ln(δ) K )-approximation algorithm for the ℓ δ,α min (Φ) metric. Finally, we show that Algorithm P performs very well even if the voters cast truncated ballots. Proposition 16 gives the relation between the number of positions used by the algorithm and the approximation ratio. In Figure 8 we show this relation for some values of the parameters m and K.
Proposition 16. Let Q be the number of top positions in the agents' preference orders that are known by the algorithm (Q ≤ mw(K) K ). Algorithm P that uses x = Q instead of Proof. Let n i denote the number of the agents not-yet-assigned until the (i + 1)-th iteration of the algorithm. Using the same reasoning as in Lemma 14 we show that n i ≤ n(1 − Q m ) i . As before, our proof proceeds by induction on i. It is evident that the hypothesis is correct for i = 0. Now, assuming that n i ≤ n(1 − Q m ) i , we assess n i+1 as follows: This proves the hypothesis. Thus, we can bound n K : This means that the satisfaction of the assignment Φ returned by our algorithm is at least: In effect, it holds that: This completes the proof.
For example, for Polish parliamentary elections (K = 460, m = 6000), it suffices that each voter ranks only 0.5% of his or her top alternatives (that is, about 30 alternatives) for the algorithm to find a solution with guaranteed satisfaction at least 90% of the one (possibly infeasible) where every voter is assigned to his or her top alternative.

ILP Formulation (Monroe, CC)
To experimentally measure the quality of our approximation algorithms, we compare the results against optimal solutions that we obtain using integer linear programs (ILPs) that solve the Monroe and Chamberlin-Courant winner determination problem. An ILP for the Monroe rule was provided by Potthoff and Brams [36], Lu and Boutilier [25] adapted it also for the Chamberlin-Courant rule with arbitraty PSF α. For the sake of completeness, below we recall the ILP whose optimal solutions correspond to α-SU-Monroe winner sets for the given election (we also indicate which constraints to drop to obtain an ILP for finding α-SU-CC winner sets): 1. For each i, 1 ≤ i ≤ n, and each j, 1 ≤ j ≤ m we have a 0/1 variable a ij indicating whether alternative a j represents agent i. For each j, 1 ≤ j ≤ m, we have a 0/1 variable x j indicating whether alternative a j is included in the set of winners.
2. Our goal is to maximize the value n i=1 α(pos i (a j ))a ij subject to the following constraints: (a) For each i and j, 1 ≤ i ≤ n, 1 ≤ j ≤ m, 0 ≤ a ij ≤ x j (alaternative a j can represent agent i only if a j belongs to the set of winners) (b) For each i, 1 ≤ i ≤ n, 1≤j≤m a ij = 1 (every agent is represented by exactly one alternative).
(c) For each j, 1 ≤ j ≤ m, x j ⌊ n K ⌋ ≤ 1≤i≤n a ij ≤ x j ⌈ n K ⌉ (each alternative either does not represent anyone or represents between ⌊ n K ⌋ and ⌈ n K ⌉ agents; if we remove these constraints then we obtain an ILP for the Chamberlin-Courant rule).

(d)
n j=1 x j ≤ K (there are exactly K winners 3 ).
We used the GLPK 4.47 package (GNU Linear Programming Kit, version 4.47) to solve these ILPs, whenever it was possible to do so in reasonable time.

Empirical Evaluation of the Algorithms
In this section we present the results of empirical evaluation of algorithms from Section 4. In the experiments we evaluated versions of the randomized algorithms that use exactly 100 sampling steps. In all cases but one, we have used Borda PSF to measure voter satisfaction. In one case, with six candidates, we have used DPSF defined through vector (3, 3, 3, 2, 1, 0) (we made this choice due to the nature of the data set used; see discussion later).
We have conducted four sets of experiments. First, we have tested all our algorithms on relatively small elections (up to 10 candidates, up to 100 agents). In this case we were able to compare the solutions provided by our algorithms with the optimal ones. (To obtain the optimal solutions, we were using the ILP formulations and the GLPK's ILP solver.) Thus we report the quality of our algorithms as the average of fractions C/C opt , where C is the satisfaction obtained by a respective algorithm and C opt is the satisfaction in the optimal solution. For each algorithm and data set, we also report the average fraction C/C ideal , where C ideal is the satisfaction that the voters would have obtained if each of them were matched to his or her most preferred alternative. In our further experiments, where we considered larger elections, we were not able to compute optimal solutions, but fraction C/C ideal gives a lower bound for C/C opt . We report this value for small elections so that we can see an example of the relation between C/C opt and C/C ideal and so that we can compare the results for small elections with the results for the larger ones. Further, for the case of Borda PSF the C/C ideal fraction has a very natural interpretation: If its value for a given solution is v, then, on the average, in this solution each voter is matched to an alternative that he or she prefers to (m − 1)v alternatives.
In our second set of experiments, we have run our algorithms on large elections (thousands of agents, hundreds of alternatives), coming either from the NetFlix data set (see below) or generated by us using one of our models. Here we reported the average fraction C/C ideal only. We have analyzed the quality of the solutions as a function of the number of agents, the number of candidates, and the relative number of winners (fraction K/m). (This last set of results is particularly interesting because in addition to measuring the quality of our algorithms, it allows one to asses the size of a committee one should seek if a given average satisfaction of agents is to be obtained).
In the third set of experiments, we have investigated the effect of submitting truncated ballots (i.e., preference orders where only some of the top alternatives are ranked). Specifically, we focused on the relation between the fraction of ranked alternatives and the approximation ratio of the algorithms. We run our experiments on relatively large instances describing agents' preferences; thus, here as in the previous set of experiments, we used Net-Flix data set and the synthetic data sets. We report the quality of the algorithms as the ratio C/C ideal .
In the fourth set of experiments we have measured running times of our algorithms and of the ILP solver. Even though all our algorithms (except for the ILP based ones) are polynomial-time, in practice some of them are too slow to be useful.

Experimental Data
For the evaluation of the algorithms we have considered both real-life preference-aggregation data and synthetic data, generated according to a number of election models. The experitments reported in this paper predate the work of Mattei and Walsh [29] on gathering a large collection of data sets with preference data, but we mention that the conference version of this paper contributed several data sets to their collection.

Real-Life Data
We have used real-life data regarding people's preference on sushi types, movies, college courses, and competitors' performance in figure-skating competitions. One of the major problems regarding real-life preference data is that either people express preferences over a very limited set of alternatives, or their preference orders are partial. To address the latter issue, for each such data set we complemented the partial orders to be total orders using the technique of Kamishima [23]. (The idea is to complete each preference order based on those reported preference orders that appear to be similar.) Some of our data sets contain a single profile, whereas the others contain multiple profiles. When preparing data for a given number m of candidates and a given number n of voters from a given data set, we used the following method: We first uniformly at random chose a profile within the data set, and then we randomly selected n voters and m candidates. We used preference orders of these n voters restricted to these m candidates.
Sushi Preferneces. We used the set of preferences regarding sushi types collected by Kamishima [23]. 4 Kamishima has collected two sets of preferences, which we call S1 and S2. Data set S1 contains complete rankings of 10 alternatives collected from 5000 voters. S2 contains partial rankings provided by 5000 voters over a set of 100 alternatives (each vote ranks 10 alternatives). We used Kamishima [23] technique to obtain total rankings.
Movie Preferences. Following Mattei et al. [28], we have used the NetFlix data set 5 of movie preferences (we call it Mv). NetFlix data set contains ratings collected from about 480 thousand distinct users regarding 18 thousand movies. The users rated movies by giving them a score between 1 (bad) and 5 (good). The set contains about 100 million ratings. We have generated 50 profiles using the following method: For each profile we have randomly selected 300 movies, picked 10000 users that ranked the highest number of the selected movies, and for each user we have extended his or her ratings to a complete preference order using the method of Kamishima [23].
Course Preferences. Each year the students at the AGH University choose courses that they would like to attend. The students are offered a choice of six courses of which they have to attend three. Thus the students are asked to give an unordered set of their three top-preferred courses and a ranking of the remaining ones (in case too many students select a course, those with the highest GPA are enrolled and the remaining ones are moved to their less-preferred courses). In this data set, which we call Cr, we have 120 voters (students) and 6 alternatives (courses). However, due to the nature of the data, instead of using Borda count PSF as the satisfaction measure, we have used the vector (3, 3, 3, 2, 1, 0). Currently this data set is available as part of PrefLib [29].
Figure Skating. This data set, which we call Sk, contains preferences of the judges over the performances in a figure-skating competitions. The data set contains 48 profiles, each describing a single competition. Each profile contains preference orders of 9 judges over about 20 participants. The competitions include European skating championships, Olympic Games, World Junior, and World Championships, all from 1998 6 . (Note that while in figure skating judges provide numerical scores, this data set is preprocessed to contain preference orders.)

Synthetic Data
For our tests, we have also used profiles generated using three well-known distributions of preference orders. Impartial Culture. Under the impartial culture model of preferences (which we denote IC), for a given set A of alternatives, each voter's preference order is drawn uniformly at random from the set of all possible total orders over A. While not very realistic, profiles generated using impartial culture model are a standard testbed of election-related algorithms.
Polya-Eggenberger Urn Model. Following McCabe-Dansted and Slinko [31] and Walsh [44], we have used the Polya-Eggenberger urn model [4] (which we denote Ur). In this model we generate votes as follows. We have a set A of m alternatives and an urn that initially contains all m! preference orders over A. To generate a vote, we simply randomly pick one from the urn (this is our generated vote), and then-to simulate correlation between voters-we return a copies of this vote to the urn. When generating an election with m candidates using the urn model, we have set the parameter a so that a m! = 0.05 (Both McCabe-Dansted and Slinko [31] and Walsh [44] call this parameter b; we mention that those authors use much higher values of b but we felt that too high a value of b leads to a much too strong correlation between votes).
Generalized Mallow's Model. We refer to this data set as Ml. Let ≻ and ≻ ′ be two preference orders over some alternative set A. Kendal-Tau distance between ≻ and ≻ ′ , denoted d K (≻, ≻ ′ ), is defined as the number of pairs of candidates x, y ∈ A such that either Under Mallow's distribution of preferences [27] we are given two parameters: A center preference order ≻ and a number φ between 0 and 1. The model says that the probability of generating preference order ≻ ′ is proportional to the value φ d K (≻,≻ ′ ) . To generate preference orders following Mallow's distribution, we use the algorithm given by Lu and Boutilier [26].
In our experiments, we have used a mixture of Mallow's models. Let A be a set of alternatives and let ℓ be a positive integer. This mixture model is parametrized by three vectors, Λ = (λ 1 , . . . , λ ℓ ) (where each λ i is between 0 and 1, and ℓ i=1 λ 1 = 1), Φ = (φ 1 , . . . , φ ℓ ) (where each φ i is a number between 0 and 1), and Π = (≻ 1 , . . . , ≻ ℓ ) (where each ≻ i is a preference order over A). To generate a vote, we pick a random integer i, 1 ≤ i ≤ ℓ (each i is chosen with probability λ i ), and then generate the vote using Mallow's model with parameters (≻ i , φ i ).
For our experiments, we have used a = 5, and we have generated vectors Λ, Φ, and Π uniformly at random.

Evaluation on Small Instances
We now present the results of our experiments on small elections. For each data set, we generated elections with the number of agents n = 100 (n = 9 for data set Sk because there are only 9 voters there) and with the number of alternatives m = 10 (m = 6 for data set Cr  Table 1: The average quality of the algorithms compared with the optimal solution (C/C opt ) for the small instances of data and for K = 3 (K = 2 for Cr); m = 10 (m = 6 for Cr); n = 100 (n = 9 for Sk).

Monroe
CC  Table 2: The average quality of the algorithms compared with the optimal solution (C/C opt ) for the small instances of data and for K = 6 (K = 4 for Cr); m = 10 (m = 6 for Cr); n = 100 (n = 9 for Sk).
because there are only 6 alternatives there) using the method described in Section 5.1.1 for the real-life data sets, and in the natural obvious way for synthetic data. For each algorithm and for each data set we ran 500 experiments on different instances for K = 3 (for the Cr data set we used K = 2) and 500 experiments for K = 6 (for Cr we set K = 4). For Algorithm C (both for Monroe and for CC) we set the parameter d, describing the number of assignment functions computed in parallel, to 15. The results (average fractions C/C opt and C/C ideal ) for K = 3 are given in Tables 1 and 3; the results for K = 6 are given in Tables 2 and 4 (they are almost identical as for K = 3). For each experiment in this section we also computed the standard deviation; it was always on the order of 0.01. The results lead to the following conclusions:  Table 3: The average quality of the algorithms compared with the simple lower bound (C/C ideal ) for the small instances of data and for K = 3 (K = 2 for Cr); m = 10 (m = 6 for Cr); n = 100 (n = 9 for Sk).  Table 4: The average quality of the algorithms compared with the simple lower bound (C/C ideal ) for the small instances of data and for K = 6 (K = 4 for Cr); m = 10 (m = 6 for Cr); n = 100 (n = 9 for Sk).
1. For the case of Monroe, already Algorithm A obtains very good results, but nonetheless Algorithms B and C improve noticeably upon Algorithm A. In particular, Algorithm C (for d = 15) obtains the highest satisfaction on all data sets and in almost all cases was able to find an optimal solution.
2. Both for Monroe and for CC, Algorithm R gives slightly worse solutions than Algorithm C.
3. The results do not seem to depend on the data sets used in the experiments (the only exception is Algorithm R for the Monroe system on data set Sk; however Sk has only 9 voters so it can be viewed as a border case).  For experiments on larger instances we needed data sets with at least n = 10000 agents. Thus we used the NetFlix data set and synthetic data. (Additionally, we run the subset of experiments (for n ≤ 5000) also for the S2 data set.) For the Monroe rule we present results for Algorithm A, Algorithm C, and Algorithm R, and for the Chamberlin-Courant rule we present results for Algorithm C and Algorithm R. We limit the set of algorithms for the sake of the clarity of the presentation. For Monroe we chose Algorithm A because it is the simplest and the fastest one, Algorithm C because it is the best generalization of Algorithm A that we were able to run in reasonable time, and Algorithm R to compare a randomized algorithm to deterministic ones. For the Chamberlin-Courant rule we chose Algorithm C because it is, intuitively, the best one, and we chose Algorithm R for the same reason as in the case of Monroe.

Evaluation on Larger Instances
First, for each data set and for each algorithm we fixed the value of m and K and for each n ranging from 1000 to 10000 with the step of 1000 we run 50 experiments. We repeated this procedure for 4 different combinations of m and K: (m = 10, K = 3),  Algorithm R for ML Algorithm C for ML Algorithm C for IC Algorithm R for IC Figure 12: The relation between the ratio K/m and the quality of the algorithms C/C ideal for the Chamberlin-Courant system; m = 100; n = 1000.
(m = 10, K = 6), (m = 100, K = 30) and (m = 100, K = 60). We measured the statistical correlation between the number of voters and the quality of the algorithms C/C ideal . The ANOVA test in most cases showed that there is no such correlation. The only exception was S2 data set, for which we obtained an almost negligible correlation. For example, for (m = 10, K = 3) Algorithm C under data set S2 for Monroe's system for n = 5000 gave C/C ideal = 0.88, while for n = 100 (in the previous section) we got C/C ideal = 0.89. Thus we conclude that in practice the number of agents has almost no influence on the quality of the results provided by our algorithms.
Next, we fixed the number of voters n = 1000 and the ratio K/m = 0.3, and for each m ranging from 30 to 300 with the step of 30 (naturally, as m changed, so did K to maintain the ratio K/m), we run 50 experiments. We repeated this procedure for K/m = 0.6. The relation between m and C/C ideal for Mv and Ur, under both the Monroe rule and the Chamberlin-Courant rule, is given in Figures 9 and 10 (the results for K/m = 0.6 look similar).
Finally, we fixed n = 1000 and m = 100, and for each K/m ranging from 0.1 and 0.5 with the step of 0.1 we run 50 experiments. The relation between the ratio K/m and the quality C/C ideal is presented in Figures 11 and 12.
For the case of Chamberlin-Courant system, increasing the size of the committee to be elected improves overall agents' satisfaction. Indeed, since there are no constraints on the number of agents matched to a given alternative, a larger committee means more opportunities to satisfy the agents. For the Monroe rule, a larger committee may lead to a lower total satisfaction. This happens if many agents like a particular alternative a lot, but only some of them can be matched to this alternative and others have to be matched to their less preferred ones. Nonetheless, we see that Algorithm C achieves C/C ideal = 0.925 even for K/m = 0.5 for the NetFlix data set.
Our conclusions from these experiments are the following. For the Monroe rule, even Algorithm A achieves very good results. However, Algorithm C consistently achieves better (indeed, almost perfect) ones. For the Chamberlin-Courant rule the randomized algorithm on some datasets performs better than the deterministic ones. However, even in such cases, the improvement over the Algorithm C is small.

Truncated ballots
The purpose of our third set of experiments was to see how our algorithm behave in practical settings with truncated ballotrs. We conducted this part of evaluation on relatively large instances, including n = 1000 agents and up to m = 100 alternatives. Thus, in this set of experiments, we used the same sets of data as in the previous subsection: the Netflix data set and the synthetic distributions. Similarly, we evaluated the same algorithms: Algorithm A, C, and R for the case of Monroe's system, and Algorithm C, and R for the case of the Chamberlin-Courant system. For each algorithm, data set, setting and each value of P we run 50 independent experiments in the following way. From a data set we sampled a sub-profile of the appropriate size n × m. We truncated this profile to the P first positions. We run the algorithm for the truncated profile and calculated the quality ratio C/C ideal . When calculating C/C ideal we assumed the worst case scenario, i.e., that the satisfaction of the agent from an alternative outside of his/her first P positions is equal to 0. In other words, we used the positional scoring function described by the following vector: m − 1, m − 2, . . . , m − P, 0, . . . 0 . Next, we averaged the values of C/C ideal over all 50 experiments.
The relation between the percentage of the known positions in the preference profile and the average quality of the algorithm for the Monroe and Chamberlin-Courant systems are plotted in Figures 13 and 14, respectively. We omit the plots for Mallow's model, as in this case we obtained almost identical results as for the Urn model. We have the following conclusions. 2. The small elections with synthetic distributions appear to be the worst case scenarioin such case we require the knowledge of about 40% of the top positions to obtain the highest approximation ratios of the algorithms. In the case of the NetFlix data set, even on small instances the deterministic algorithms require only about 8% of the top positions to get their best quality (however the quality is already high for 3-5% of the top positions). For the larger number of the alternatives, the algorithms do not require more than 3% of the top positions to reach their top results.
3. Algorithm C does not only give the best quality but it is also most immune to the lack of knowledge. These results are more evident for the case of the Monroe system.

Running time
In our final set of experiments, we have measured running times of our algorithms on the data set Mv. We have used a machine with Intel Pentium Dual T2310 1.46GHz processor    Figure 15 we show the running times of the GLPK ILP solver for the Monroe and for Chamberlin-Courant rules. These running times are already large for small instances and they are increasing exponentially with the number of voters. For the Monroe rule, even for K = 9, m = 30, n = 100 some of the experiments timed out after 1 hour, and for K = 9, m = 30, n = 200 none of the experiments finished within one day. Thus we conclude that the real application of the ILP-based algorithm is very limited. Example running times of the other algorithms for some combinations of n, m, and K are presented in Table 5. For the case of CC, essentially all the algorithms are very fast and the quality of computed solutions is the main criterion in choosing among them. For the case of Monroe, the situation is more complicated. While for small elections all the algorithms are practical, for elections with thousands of voters, using Algorithm GM becomes problematic. Indeed, even Algorithm C can be seen as a bit too slow if one expects immediate results. On the other hand, Algorithms A and B seem perfectly practical and, as we have seen in the previous experiments, give high-quality results.

Summary
We have defined a certain resource allocation problem and have shown that it generalizes the problem of finding winners for the multiwinner voting rules of Monroe and of Chamberlin and Courant. Since it is known that the winners for these voting rules are hard to compute [7,25,37,42,43], we focused on finding approximate solutions. We have shown that if we try to optimize agents' dissatisfaction, then our problems are hard to approximate up to any constant factor. The same holds for the case where we focus on the satisfaction of the least satisfied agent. However, for the case of optimizing total satisfaction, we suggest good approximation algorithms. In particular, for the Monroe system we suggest a randomized algorithm that for the Borda score achieves an approximation ratio arbitrarily close to 0.715 (and much better in many real-life settings), and (1 − 1 e )-approximation algorithm for arbitrary positional scoring function. For the Chamberlin-Courant system, we have shown a polynomial-time approximation scheme (PTAS).
In Table 6 we present the summary of our (in)approximability results. In Table 7 we present specific results regarding our approximation algorithms for the utilitarian satisfaction-based framework. In particular, the table clearly shows that for the case of Monroe, Algorithms B and C are not much slower than Algorithm A but offer a chance of improved peformance. Algorithm GM is intuitively even more appealing, but achieves this at the cost of high time complexity. For the case of Chamberlin-Courant rule, theoretical results suggest using Algorithm P (however, see below).
We have provided experimental evaluation of the algorithms for computing the winner sets both for the Monroe and Chamberlin-Courant rules . While finding solutions for these rules is computationally hard in the worst case, it turned out that in practice we can obtain very high quality solutions using very simple algorithms. Indeed, both for the Monroe and Chamberlin-Courant rules we recommend using Algorithm C (or Algorithm A on very large Monroe elections). Our experimental evaluation confirms that the algorithms work very well in case of truncated ballots. We believe that our results mean that (approximations of) the Monroe and Chamberlin-Courant rules can be used in practice.
Our work leads to a number of further research directions. First, it would be very interesting to find a better upper bound on the quality of solutions for the (satisfactionbased) Monroe and Chamberlin-Courant systems (with Borda PSF) than the simple n(m −  In column "Approximation" we give currently known approximation ratio for the algorithm under Borda PSF, on profiles with m candidates and where the goal is to select a committee of size K. Here, O(Φ S ) = O(n 2 (K +logn)) is the complexity of finding a partial representation function with the algorithm of Betzler et al. [7]. w(·) denotes Lambert's W-Function.
1) bound that we use (where n is the number of voters and m is the number of candidates). We use a different approach in our randomized algorithm, but it would be much more interesting to find a deterministic algorithm that beats the approximation ratios of our algorithms. One of the ways of seeking such a bound would be to consider Monroe's rule with "exponential" Borda PSF, that is, with PSF of the form, e.g., (2 m−1 , 2 m−2 , . . . , 1). For such PSF our approach in the proof of Lemma 9 would not give satisfactory results and so one would be forced to seek different attacks. In a similar vein, it would be interesting to find out if there is a PTAS for Monroe's system. In our work, we have focused on PSFs that are strictly increasing/decreasing. It would also be interesting to study PSFs which increase/decrease but not strictly, that is allowing some equalities. We have started to work in this direction by considering the so-called tapproval PSF's α t , which (in the satisfaction-based variant) are defined as follows: α t (i) = 1 if i ≤ t and otherwise α t (i) = 0. Results for this case for the Chamberlin-Courant rule are presented in the paper of Skowron and Faliszewski [42].
On a more practical side, it would be interesting to develop our study of truncated ballots. Our results show that we can obtain very high approximation ratios even when voters rank only relatively few of their top candidates. For example, to achieve 90% approximation ratio for the satisfaction-based Monroe system in Polish parliamentary election (K = 460, m = 6000), each voter should rank about 8.7% of his or her most-preferred candidates. However, this is still over 500 candidates. It is unrealistic to expect that the voters would be willing to rank this many candidates. Thus, how should one organize Monroebased elections in practice, to balance the amount of effort required from the voters and the quality of the results? Finally, going back to our general resource allocation problem, we note that we do not have any positive results for it (the negative results, of course, carry over from the more restrictive settings). Is it possible to obtain some good approximation algorithm for the resource allocation problem (in the utilitarian satisfaction-based setting) in full generality?