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The Minimum Maximal Independent Set problem is, given a parameter k, find if thereexists a maximal independent set of size at most k. For Minimum Maximal Independent Setwe are able to show that unless P = NP, for any increasing function g, h, there is no g(opt)approximation for the problem in time h(opt) for any g, h, namely, we are able to prove theFellows conjecture for MMIS. This seems to be the only non trivial problem for which we canprove such a result so far.
Our paper is motivated by an important conjecture by Fellows. The conjecture concerns parameterized approximation for Set Cover and Clique. We use a convention in fixed parameter tractabilityfor maximization problems: a solution is not admisable unless its of size w(1) (namely, superconstant). Otherwise, by returning a single vertex we approximate Clique within opt.Given this, a version of the Fellows conjecture can be described as follows:Conjecture 1.1. The Fellows conjecture [6]:For any pair of increasing functions h, g, Clique and Set-Cover admit no g(opt) approximationthat runs in h(opt) nO(1)We are only interested in time h(opt) that is super exponential in opt. We try to avoid asituation that hardness results are directly translated to FPT inapproximability. Indeed, if h may besubexponential then in many cases a polynomial gap reductions will imply FPT inapproximability.In any polynomial gap reduction from 3-SAT, an instance I of 3-SAT is mapped to an instanceof size |I|D of some problem, for a constant D > 1. In addition, many times opt |I|D holds.1/D01/D0Then for any D0 > D, 2opt= 2o(|I|) . Thus setting h(opt) = 2opt , is a direct way to get FPTinapproximability from a regular inapproximability result. However, if we insist that h(opt) will besuper exponential in opt, proving FPT inapproximability, is equivalent to creating a gap reductionwith very small optimum. This is by no means an easy task. The two inapproximability resultsfor Clique [10] and for Set Cover [17] seem hard to use, if a super exponential time is required.The reductions from an instance I of 3-SAT to Clique and Set-Cover derive an optimum of sizequite close to |I|D for D > 1. Thus even exponential time in opt is out of the question, unless wetransform the instance to a different instance with smaller opt. A thing that seems highly difficultto do. To overcome this difficulty, we use other inapproximability results. Proving the Fellowsconjecture seems beyond the reach of current knowledge. Thus we try to prove such results forlarge g and h.Another problem we study is the Minimum Maximal Independent Set problem that seeks aminimum size independent set that is also maximal (namely is a dominating set). This problemseems quite hard to approximate within better than n. For every feasible set S, every super set of Sand every subset of S may not be feasible. The problem is known to be W[2]-Hard [7] and unlessP = NP it admits no n1 approximation [9]. There are very few problems we are aware of that
admit such a strong hardness without using any PCP theorem. Another example is a beautiful ninapproximability for the maximum number of disjoint paths in a directed graph [8].Remark: In all future calculation we assume that all numbers are integral. Fixing this usingfloors and ceilings is elementary.
Previous work
The following relation is known among the parameterized complexity classes: FPT W [1] W [2]. We are going to assume the ETH and thus use the fact that it is known that under ETHClique and Set-Cover do not have an FPT when parameterized by the size of the solution.Corollary 2.1. (FPT6= W[1]), i.e., C LIQUE cannot be solved in time f (k)nc where f is any functiondepending only on k, and n is the number of vertices of the graph G.Corollary 2.1 implies the following weaker claim:Corollary 2.2. (FPT6= W[2]), i.e., S ET C OVER cannot be solved in time f (k)nc where f is anyfunction depending only on k, and n is the size of the universe.In proving FPT for a problem, there is always a parameter k. We are only interested in k thatis the value of the optimum. For inapproximability we set k = opt which is a more strict inapproximability notion. In the parameterized by k inapproximability for a minimization problems,the goal is to show that there is no g(k) ratio in time h(k)nO(1) , for some k opt (in maximizationwe impose k opt). Setting k = opt (namely, k is the optimum of some concrete instance) is amore strict notion than showing inapproximability of some k opt. Hence we adopt the latterstricter version. We note that opt is the optimum of some instance and as we shall see it will be theoptimum of a yes instance in gap reductions from 3-SAT.To the best of our knowledge, the effort of showing FPT inapproximability for Clique and SetCover started with [2].Theorem 2.3. [2] Under the ETH and PGC, there exist constants F1, F2 > 0 such that the SetFCover problem does not admit a FPT approximation algorithm with ratio opt F1 in time 2opt 2 .The above theorem uses F2 < 1 hence uses time subexponential time opt and is not suited forthis paper.Theorem 2.4. Unless NP SUBEXP, for every 0 < < 1 there exists a constant F = F( ) > 0Fsuch that Clique admits no FPT approximation within opt 1 , in time 2opt .As F < 1 in the above construction the running time here two is subexponential in opt and thistheorem is not suited for this paper.We note that the price of wanting super exponential time is getting much weaker inapproximability result than [2]. however, it is our belief that the definition of FPT inapproximability shoulddemand super exponential running time in opt.In [2] a large collection of W [1]-hard problem for which Fellows conjecture does not apply,is presented, namely, some problems are given f (opt) ratio. And in fact, the running time is justpolynomial in the size of the input. All these problems are not only W [1]-hard but also admits astrong inapproximability results (at least Label-Cover hardness. See [15]) Thus for some reasonsome W [1]-hard problems seem to behave different than Clique.Theorem 2.5. [2] Directed Multicut, Directed Steiner Tree, Directed Steiner Forest, DirectedSteiner Network and Minimum Size Edge Cover admit g(opt)-approximation algorithms for somesmall function g (the largest approximation ratio we give is opt 2 ). The running time is polynomialin nTheorem 2.6. [2] Strongly Connected Steiner Subgraph problem (which is W[1]-hard and does nothave any polynomial time constant factor approximation) admits a 2 factor h(opt)nO(1) algorithm.3
This seems to be the only natural problem for which such a result is known.Previous work from inapproximability theory:Theorem 2.7. [10, 18] Unless P = NP, Clique can not be approximated within n1 .The [10] reduction was randomized but in [18] this result is derandomized and it is achievedunder the assumption that P 6= NP. Note that there are stronger inapproximability results for Clique[13] but we can not use them because the running time of this reduction is quasi polynomial (infact if the reductions would have been polynomial, this would contradict the ETH).Theorem 2.8. [17] Unless P = NP Set Cover admits no better than c ln n approximation for someconstant c.These inapproximability results are close to best possible. Clique admits a a trivial n ratioapproximation. For Set Cover it is known (folklore) that the natural greedy algorithm gives aln n + 1 approximation.
This paper illustrates the importance of linear reductions. There is a linear reduction from aninstance I of 3-Sat to Clique. More precisely, the number of edges in the graph is not linear in |I|,but the number of vertices is linear in I. This is enough for proving our result for Clique.To give hardness to Set Cover we use an almost linear PCP (see [15]). 1 Say that a PCP has size|I| (n) for (n) = o(|I|). Our technique changes the optimum to (n), hence making it smaller.There are standard reduction from almost linear PCP [15], to Set Cover for example the one of[14]. Note that the number of elements in the Set Cover reduction is not almost linear in |I| butrather polynomial in |I| (as we shall see). But for our result, its enough that the number of sets isalmost linear in |I|.In [15] its is conjectures that there exists a PCP of size |I| (I) poly(1/), with gap 1/ andalphabet of size poly(1/), for (I) = o(|I|). By a proper choice of , and the use of our technique,we turn the optimum to (|I|) = o(|I|) which is enough to guarantee time super exponential in opt.
In this section, we briefly describe the complexity conjectures assumed in this paper.
Impagliazzo et al. [11] formulated the following conjecture which is known as the ExponentialTime Hypothesis (ETH).E XPONENTIAL T IME H YPOTHESIS (ETH)3-SAT cannot be solved in 2o(n) (n + m)O(1) time where n is the number of variables and mis the number of clauses.Using the Sparsification Lemma of Calabro et al. [1], the following lemma was shown.Lemma 4.1. Assuming ETH, 3-SAT cannot be solved in 2o(m) (n + m)O(1) time where n is thenumber of variables and m is the number of clauses.1 Its
not hard to see that if there is a reduction from 3-SAT to Set-Cover with linear number of sets, the ETH fails
The main conjecture used in this paper from the world of approximation algorithms is the Projection Games Conjecture due to Moshkovitz [15].P ROJECTION G AMES C ONJECTURE (PGC)There exists c > 0, such that for every n1c , it is possible to design an almost-linear sizePCP Theorem with completeness 1 and soundness and alphabet size poly( 1 ).Moshkovitz is able to show that under PGC there is no (1 ) ln n-approximation for Set Coverunless P = NP. The PCP of [15] is in fact a Labelcover instance [15]. In other words, the proverqueries two entries of the PCP string. The following can be obtained from the PGC using standardreductions (see [14]).Conjecture 4.2. [15] There exists a reduction from a SAT instance of size I to an instance of set
cover with at most n = |I| 2log (|I|) poly( 1 ) sets so that the gap between a yes and a no instanceis ( 1 )While the almost linear term of the [15] conjecture is not specified, we did the natural thing;We used the almost linear expression of the paper [16], which is last PCP paper by Moshkovitzbefore [15].The conjecture of Fellows is:Conjecture 4.3. For any two increasing functions f , g, Clique and Set Cover admit no g(opt)approximation in time h(opt)nO(1) .
S ET C OVERInput : A universeU = {u1 , u2 , . . . , un } and a collection S = {S1 , S2 , . . . , Sm } of subsets ofSmU such that j=1 S j = U.SProblem: Does there exists a subcollection S 0 S of size k such that Si S 0 Si = U?Parameter: kM INIMUM S IZE M AXIMAL I NDEPENDENT S ET:Input : An undirected graph G = (V, E), and an integer kProblem: Does G have a maximal independent set of size at most k?Parameter: k
Our Results
Theorem 5.1. Under the ETH and the PGC conjectures, the exists two constants f , > 0 so thatfSet cover is (log1+ (opt)) inapproximable, even in time exp((opt (log opt) )) for some constantf.This time much larger than exponential in opt and so meets the main requirement of the paper.Theorem 5.2. If there exists a PCP of size |I| polylog|I| log(1/) and gap 1/ then set cover0d 00admits no opt d ratio, for some constant d 0 , even in time exp(2opt ) for some constant d 00 .This PCP, was conjecture to exist, in a private communication, by Dana Moshkovitz, who is aleading expert in the subject of PCP. Note that the running times is almost doubly exponential inopt.Theorem 5.3. Under the ETH there exists a constant so that for every function h, Clique admitsno 1 approximation, even in h(opt)n(O(1) timeThe above theorem is proved by creating an instance of Clique with constant gap, so that theoptimum size clique can be f (opt) for any f , however small. This seems to be a new construction.We can compare our result to the one of [5]. In [5] it is shown that for opt log n, Clique cannot be solved in significantly smaller time than nopt time, unless NP has subexponential simulation.More specifically, if Clique can be solved in much smaller time than nopt , any solution for an NPCproblemp that makes f (n) non deterministic steps, implies a deterministic solution in time roughlyexp( f (n)). This implies that NP admits a subexponential algorithm.Theorem 5.3 works for any opt and opt log n in particular. Thus for opt log n Clique hasno 1 inapproximability in time h(opt)nO(1) for any increasing function h. The improvementover [5] is that we show inapproximability and not rule out exact solutions. For such small valuesof opt it may well be that getting an exact solution is considerably harder than getting 1 approximation. However, our improvement is not strict in the following sense. The times in [5] andour paper are non comparable, because [5] allows slightly super polynomial time while we allowh(opt) poly(n) time for any function h. It may be the case that combining the ideas of this paperand [5] would give a running time which is h(opt) times the running time of [5].Remark: We prove our Clique inapproximability under the ETH while in the [5] an exact solution is ruled out under a somewhat stronger assumption, namely, that NP admits no subexponentialsimulation.We find a well known problem for which we are able to show that the Fellows conjecture holdtrue.Theorem 5.4. For any two increasing functions g, h, MMIS admits no g(opt) ratio in time h(opt) nO(1)Except such a trivial result for 3-coloring (see below) that is the only problem for which we areable to prove Fellows conjecture on, so far. We will discuss in details the special properties of the(simple) reduction from 3-SAT to MMIS, and why does it allow proving the conjecture.In the rest of the paper we will assume that all numbers are integers. This is done to simplifythe claims. An elementary use of ceiling and floors can correct this point.Avoiding a constant optimum: Consider the 3-Coloring problem, We (trivially) show that theFellows conjecture holds for this problem. In [4] it is shown that 3-coloring admits no constant6
approximation for any constant unless a variant of the (well known) Khot 2 to 1 projection gameconjecture holds. Take the yes instance of the problem (in which opt = 3). For any g, g(opt) = g(3)is constant and h(opt) nO(1) is just polynomial time. By the above hardness of [4], it is clear thatthe Fellows conjecture applies for 3-coloring. The lesson to be derived from this example is thatwe should only try to prove the Fellows conjecture for problems with non constant opt.
We choose = c/ ln2 n for a large enough constant c for Conjecture 4.2. (interestingly, in [15] achoice of = c/ log4 n is required. We however can afford to choose slightly larger ).Thus the PGC implies:Corollary 6.1. (A variant of [14] applied on [15]) There exists a reduction from the Projection
the optimum is |I| 2log |I| polylog(|I|) and case of a no instance, the solution size is c ln |I| |I|
from [16]. This length is |I| 2log |I| polylog(|I|)(1/)c1 for some constant c1 . The PCP of [15]is equivalent to a Labelcover instance. In Labelcover, we have two provers and a collection ofquestions for the two provers. The total number of questions is the size of the PCP. Two questionsthat can be asked together is called a query. The verifier asks a query. After getting the answersfrom both provers, the verifier performs a consistency test and answers yes if this test is successful,and no otherwise.Note that the alphabet et size in [15] is (1/)c2 for some constant c2 and so clearly the numberof answers per query can not be more than that. The size the Label-Cover instance (for both yesand no instances) is the number of questions times the number of answers per question and so is:
|I| 2log |I| polylog(|I|)(1/)c1 +c2 . The o(|I|) term is taken from [16]. The choice of allows us
to write the size of the Labelcover instance as |I| 2log |I| polylog(|I|)(1/)c2 .It is always the case that for a yes instance we can choose a single answer to every question
and satisfy all queries (see both [14] and [15]). Thus for a yes instance we can pick |I| 2log |I| polylog(|I|) answers and satisfy all queries.Given any strategy (single answer per question), for a no instance the prover accepts for at most1/ log2 n of the queries (the fraction of queries that is accepted is ). This 1/ gap combined witha gadget inpwhich every answer for a prover is a set (see [14]) gives a gap reduction for Set Coverwith gap 1/ = (log |I|). The size of the optimum in a yes instance for Set Cover is the number
of answers to satisfy all questions namely, |I| 2log |I| polylog(|I|). For a no instance the optimumis (log |I|) larger.Note that in [14] a ground set of elements is defined for any query. A collection of partitionsis deterministically chosen per ground set of elements. There is a bijection between partitions andsets. Even though the construction is deterministic, to understand this better one can think of everypartition as a random half of the ground set of elements. The size of every ground set of elementsis M = (2dlog n ) for some constant d for reasons to be clear soon. For a yes instance there is atrap door namely its possible to cover any ground set of elements with 2 sets (partitions) namely,7
a partition and its complement. But in a no instance we need to use about ln M = (log(|I|) sets(partitions) to cover the ground set (just as covering a set with random partition). This leads toln n/2 inapproximability. The size of the ground set is M =(2dlog n ) for some constant d becauseas we saw the gap for Set Cover is roughly min{log M, 1/ }, which by the choices of M and itis c ln n for some constant c.Thus the number of elements is indeed bounded by |I|3 poly(n) because the number of queriesis less than |I|3 .
We now describe a way to change the Set Cover instance to a new one. The idea is to make theoptimum much smaller. Let the set cover instance be G(X,Y, E) with X the sets, Y the elements.The set E is a simple way to describe which element belong to which set. In particular, there is anedge between a set and an element, if the element belongs to the set.We are interested in large subsets of X. Select all subsets of size |I|/(log(|I|) of X. Such abig set will be called a collection set to distinguish between these super sets and the sets of theinstance we start with. Let S be a typical collectionSset, S = {s1 , s2 , . . .} with every si a set, namely,a collection of elements. Connect every such S to sS s namely, to all the elements that belong tosome s S.This defines a new Set-Cover instance with the collection sets as the sets, and the sameelements. The inclusion of elements in a collection set is as described above.Claim 6.2. The size of the new graph is 2o(|I|) as required by the ETHProof. The number of collection sets is
|I| 2log |I| polylog(|I|)
|I|log |I| .
We use
nkk (ne/k)
On the other hand the size of the new optimum for a no instance isX c ln n,|I|/ log(I)proving the claim. 2 |I|
Claim 6.4. The new optimum value for a yes instance is 2log
Proof. As discussed above we should divide |I| 2log |I| polylog(I) for < 1 by |I|/log(|I|). Theclaim follows as an additional log |I| terms is swallowed by the polyog(|I|) term. 2Since the gap is ln |I| up to constants we get:Claim 6.5. The gap is larger than (log opt)1+ for some constant > 0.
Proof. The new optimum is 2log |I| polylog(I). Thus log opt = O(log |I|). Thus for any constant > 0 so that 1 + < 1/ we get:((log opt)1+ ) = o(log |I|)Which implies that the gap is actually larger than (log opt)1+ .
and is 2o(|I|) if d is large enough. This is implied by the inequality nk (ne/k)k . The reason for
Our technique is very specialized, and this technique alone can not be used to prove the Fellowsconjecture for Set-Cover. Only a linear reduction from 3-SAT to Set Cover can be used to provethe conjecture. However such a reduction can not exist as it contradicts the ETH.There is a large evidence that a linear PCP can not exist. The ultimate PCP we may expect(albeit this is not known even for constant ) is a PCP of size |I| poly(1/) with gap 1/. Forthe choice of 1/ = polylog(n) the inapproximability is almost the same as in Theorem 5.2. Thisshows that Theorem 5.2 got the best result possible if we only use almost linear PCP and ourtechnique.Other methods: It is plausible that we can make the optimum smaller by random samplingtechniques, and combine it with the above technique of making collection sets, we would be ableto prove the Fellows conjecture. However at this stage we were not able to achieve that. It maybe the case that the current PCP theory does not allow the proof of the Fellows conjecture for SetCover. In fact we suspect that a new type of PCP, namely, a parameterized version of the PCPtheorem has to be invented for proving the Fellow conjecture, at least for Set-Cover.
.We use the elementary reduction from 3 SAT with a constant gap to the Clique problem. Inthis case the size of instance I of the 3-SAT is denoted by |I| = m +n with n the number of variablesand m the number of clauses.10
In the reduction, we add 7 vertices for every fixed clause in the 3-SAT instance. One per each 7satisfying assignments to the clause. Thus we get a correspondence: a vertex in the clique graph isa legal assignment to some clause. We join two vertices (assignments for two clauses) by an edgeif the assignments do not contradict. This can be a fake edges namely edges from assignment to C1to an assignment C2 so that C1 and C2 share no variables. However, if C1 and C2 share a literal or avariable, the two assignment are joined by an edge only if the truth value for the mutual variable(s)is the same. These edges are called true edges. Clearly the seven possible assignments for a fixedclause C are an independent set, because they disagree (by definition) on the value of at least onevariable.Theorem 7.1. The standard PCP theorem: There exists a reduction from any NPC language Lto 3 SAT with n variables and m clauses so that a yes instance is mapped into a 3-SAT instancesuch that all the m clauses can be simultaneously satisfied. In the case of a no instance, at most1 fraction of the clauses can be simultaneously satisfied.The following simple claim is standard and is proved for completeness.Claim 7.2. If the 3-SAT is a yes instance there is a clique of size m with m the number of clauses.For a no instance, the maximum clique set is of size at most (1 )mProof. In case of a yes instance there is a truth assignment F that satisfies all clauses. Fix a clauseC. Among the 7 possible truth assignment per clause, we take the one that agrees with F. Theclaim for a yes instance follows. For a no instance say for the sake of contradiction that the cliqueis of size larger than (1 )m. Ignore the fake edges. Recall that the 7 assignments to the sameclause are an independent set, because by definition they disagree on some variable. Hence therecould be at most one vertex (assignment of values for the variables of the clause) for every clause.In particular, there is a one-to-one function from vertices in the clique to clauses.The clique of size larger than (1 )m corresponds to a collection of clauses larger than (1 )m because the above assignment is one-to-one. However, this gives a collection of more than(1 )m clauses that can be simultaneously satisfied for a no instance. Indeed, every two clausesthat share a variable assign the same value to the variable because these two vertices are a part ofa clique. This is a contradiction. 2We do the following transformation that is a modification of what we did for for Set-Cover.The number of vertices is 7m. Let f (m) be any (slowly) increasing function of m. We make everysubset of m/ f (m) vertices of the 7m vertices a single vertex. Such a vertex is called a collectionvertex. Two collection vertices A, B, are connected by an edge, if A B is a clique, and even moreimportantly A B = 0./ The last condition, namely, the fact that two sets that are connected mustbe disjoint is not needed in the Set Cover reduction, but it is crucial here.Claim 7.3. The new instance of the Clique problem has size 2o(m)Proof. Using nne k
,kkwe get that the number of collection vertices sets is at most (7 e f (m))7m/ f (m) . Note that this isalso equal to: 2log(7e f (m))7m/ f (m) . The number of edges in the new Clique instant is at most thenumber of vertices squared hence at most 22log(7e f (m))7m/ f (m) which is at most 2o(m) because f isan increasing function. This size is within the limitation of the ETH (because of the Sparsificationlemma). 211
Do the following for every clause C. For every variable xi , if xi appears in C add an edgebetween vi to all vertices in W (C). Otherwise, if xi C add edges from vi to all of the vertices ofW (C).We now discuss the properties of this reduction. Note that when we say opt it is the value ofthe optimum of a yes instance in some gap reduction.We note that if the formula is satisfiable, we can take a subset of vertices that corresponds to asatisfying assignment. This is the union of two sets QY {vi } and QN {vi } so that for every x j ,either v j Qy or v j QN . Namely, the subset are defined according to a satisfying assignment. Asthe assignment is legal, the union QY QN will not contain an edge because every xi gets either aT value or a F value but never both. In addition, every clause vertex will be covered by the vertexthat satisfies the clause. More precisely, for every clause C there exists some xi so that eitherxi C QY or xi C QN . The corresponding vertex vi or vi will be a neighbor of all the copiesof C. Thus the size of the optimum is n and is linear in the number of variables.As we shall see, the optimum for a no instance not linear in the 3-SAT instance. But the factthat the optimum for a yes instance is linear in the number of variables, (even if its not linear inany other sense) turns out to be enough.We now do the usual trick for reducing the value of the optimum in the case of a yes instance.For some arbitrarily small f (n) we look at every set of size n/ f (n) that does not contain both viand vi as a super vertex. Two super vertices are joined by an edge if their union corresponds to anindependent set. A super vertices is said to contain the vertices he is associated with.The edges of supervertices are defined as follows. A super vertex is joined to the union of allneighbors of the vertices that belong to the super vertex. By the analysis we have already seen, thisreduces the optimum in the case of a yes instance to f (n) as we can take a collection of verticesthat corresponds to a satisfying assignment and split it to f (n) disjoint sets of size n/ f (n) each.Also, we have seen (when we prove the result for Clique) that the size of the construction is 2o(n) .For a no instance, regardless of which collection of super vertices is taken, we must get anindependent set. This means that the collection of super vertices chosen, in case of a no instance,is a (maybe partial) legal assignment of values to the variables (in the usual sense, namely if asuper vertex containing vi is chosen, then we assign T to xi ).However, the truth assignment for a no instance can not satisfy all clauses. This means that forsome C, C is not connected to any of the super vertices chosen by the optimum for the no instance.Since every C corresponds to q(n) vertices the size of the optimum in a no instance, regardlessof the super vertices chosen, is at least q(n) (all the copies of the non satisfied clause have to beadded to the independent set, for otherwise it will not be a dominating set).Thus the situation is like in Claim 8.1 namely, we can make the optimum f (n) for as small aswe want function, and the value of a no instance is at least q(n) for some increasing function q(n).This implies that the Fellows conjecture holds for MMIS.
Open problems
To disprove Fellows conjecture for a problem we need to give a g(opt) approximation for a problem that runs in h(opt)nO(1) time. This question was already systematically addressed in [2]. Alarge collection of W[1]-hard that also have a strong inapproximability are given g(opt) approximation for some increasing function g. The algorithms of [2] run in polynomial time. However, allproblems presented were only W[1]-hard.Open question 1: Is there an f (opt) approximation for any W[2]-hard problem, that runs inpolynomial time?13
It may be hard to prove that such a thing does not exist as it will imply that W[1]6=W[2]. Still,this suggests an indirect way to try and prove this important and widely believed conjecture.Another way to make the optimum smaller is by taking a random sample.Open question 2: Can random sampling drastically reduce the size of the optimum in certainproblems? In particular, is it possible for Clique?
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[15] D. Moshkovitz. The Projection Games Conjecture and The NP-Hardness of ln nApproximating Set-Cover. APPROX, 2012.[16] D. Moshkovitz and R. Raz. Two-query pcp with subconstant error. J. ACM, 57(5), 2010.[17] R. Raz and S. Safra. A sub-constant error-probability low-degree test, and a sub-constanterror-probability pcp characterization of np. In STOC, pages 475484, 1997.[18] D. Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3(1):103128, 2007.


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Finest Greek Tastes is an ambitious LEADER cooperation project involving 21 Local Action Groups and more than 50 municipalities, working together to promote quality Greek products based on local culinary traditions and culture.
Projects & Practice
A French region developed its own brand that capitalises on its culinary tradition, as well as the diversity and quality of its local produce.
The project supported the area of Chimay to promote 'green tourism' and boost local development by fostering cooperation among local stakeholders.
An innovative LEADER project promotes hiking in the Belgian Eifel area by means of an online hiking planner tool, GPS navigation and a smartphone app.
Awareness raising and sales promotion for organic crops and products
The "Living on the Edge" route comprises of 21 different sites presenting the nature, culture and history of South-Estonia using the yellow frame, the iconic symbol of "National Geographic".
An Estonian NRN TV-project helped change the public perception about rural life.

subset: commoncrawl


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We hired Harvey's to paint the exterior of our house. We had to wait a bit to have the work done, but it was worth it! They did a fantastic job! They showed up when they said they would and finished as planned. When they were finished, they cleaned up everything. We didn't have to pick up anything! I had other bids for the same job and Harvey's price was more than fair and the quality was top-notch! After it was completed, they called to make sure everything was to our satisfaction and told me to call them if we found any problems or areas needing touch ups. I highly recommend Harvey's House Painting!
So sad that Angie's List requires the contractor be notified!! They have lost me as a client. If I'm dissatisfied and want to give an "F" rating (question/workers...I SHOULD NOT be intimidated into passing up an opportunity to explain my experience! There is NO WAY my contractor got an Aplus rating from over five hundred people....so someone was lying....yet I can't report that without the contractor knowing it? REALLY?
House Painting Highlands Ranch Colorado 80163
"Service beyond Expectation" is our motto. I have 40 years experience in painting and the building/remodeling trades. I exclusively use Sherwin Williams, and am well versed in their products and utilize this to meet my customers needs. I am a small business with 6-10 employees. Service and dependability are my passion, and I strive to form relationships with all my customers.
Tru Colors Contracting is an experienced residential paint contractor that has been serving Miami, Broward and Palm Beach Counties for many years. We are completely dedicated to providing our residential customers with high-quality interior and exterior painting services that yield beautiful and long-lasting results. Painting is one of the easiest ways you can enhance the look of your home and that's what we will do for you with our services. We can paint the inside and outside of your home, including any structure or surface that is looking dull or worn and needs a new look.
Meet the pros. Call in at least three different contractors for your job. (Friends and family are good reference sources.) And be home for the initial meeting with them. That way, you'll know how much time each contractor took to assess the condition of your home. The longer he takes, the more realistic the estimate you'll get. Even an experienced painter will need more than a quick walk around your house.
the price can always be adjusted based on on or off season, amount of colors to be used. A job like this can be done in 2 days and much quicker with a helper. I charge a bit more than others just because its quality work done professionally. Theres others that charge more than this just because they can on the on season and most people dont get a 2nd bid...
I've seen this article sent out 4 times now. It's out dated in terms of today's "Scams." Paint stores haven't charged more money for deeper paint colors in over 15 years. Also Water downed paint? Seriously? That would never happen in the real world. And Wall repair? what painter would ever not assume wall repair as included? This article is a waste of time and should be taken down or edited to reflect actual scams of todays times.
The cost to paint a room will vary based on the size of the room, the type of paint you select, any ceiling or trim work needed, and any repairs needed. If no repairs or other additional work is needed, painting a room with contractor-grade paint could range from $1 to $2 per square foot, including materials. Selecting a luxury paint (some cost more than $100 per gallon) will certainly increase your house painting costs. Here are some additional examples of average house painting costs, per room:
We are completely dedicated to making sure our customers are satisfied with the painting services we provide. We use only our top notch tools and paint so you know you can depend on the work we do for you. We will complete any job you hire us for as efficiently as possible and within any budget you set. As a residential paint contractor, we can paint the interior or exterior of your home or office. We can also provide our pressure cleaning service on any surface on your property to get rid of dirt and make sure it's completely clean and ready for us to paint.
We pride ourselves on providing the best possible customer service - from the first contact to the completion of the job. The job is not finished until you are completely satisfied with our work. With a variety of paints, colors, and services to choose from, we are sure you will be happy working with us. Look around our website and if you have any questions, please feel free to contact us. We hope to see soon!
To win work, contractors generally have to bid against other suppliers. To make an accurate bid, they estimate the time and materials required to complete the job. They measure the area to be painted and discuss the type of finish the customer requires. They assess the quality of the surfaces to work out how much remedial work they must carry out before painting. They might have to remove old paint or wall coverings, or repair cracks and other damage. They also calculate the cost of any essential equipment, such as scaffolding for exterior walls.
I have a Home Improvement/Painting business, and Angie's List always advertises that that everyone is out to get them. Of course there are people who try to take advantage of homeowners My reputation and repeat business is based on word of mouth. Shoddy work is always a way to get put out of business quick. As far as strictly painting, preparation is a big factor in getting a quality paint job. If you don't prepare the surfaces you are painting you are spinning your wheels, and wasting money, no matter what paint you use. Getting a deposit from a customer is beneficial, but not always necessary. Sometimes it is a godsend, when you get stuck by the customer, which has happened to me more than once House Painting Home Painters Highlands Ranch Colorado Click Here
Meet the pros. Call in at least three different contractors for your job. (Friends and family are good reference sources.) And be home for the initial meeting with them. That way, you'll know how much time each contractor took to assess the condition of your home. The longer he takes, the more realistic the estimate you'll get. Even an experienced painter will need more than a quick walk around your house.
At Supreme Painting, you can feel good about hiring professional painters. An Exceptional Customer Experience is our #1 Priority and we get the job done right and as scheduled. Your painting project will be done quickly with as little impact on your daily routine as possible. We respect your time. Other than the beautiful new look of your space, you won't even know that we've been there.
The other difficult part is getting a painting contractor to show up. While this generalization does not apply to every painter, personally I am extremely grateful if I can get a paint contractor to show up to look at the house and to later produce a written estimate. I hardly fault the painting contractors, because I think it is a combination of the contractors being smaller operations along with a high demand for their work.
Hello, I have a sad situation to share -- a friend of mine who is a very good painter, experienced too, fell off a tall ladder that did not have "boots" on it. (I've never seen those.) Anyway, do you think he should have asked for boots before painting? Possibly it was a situation where he was shy to ask because he wanted the job... (I don't know all the details.)
The obvious trick here is for unaffiliated contractors to use the BBB, PDCA, Chamber or other organizations' logos without being a member. Other tricks include claiming membership when that membership has expired or even make up fake organizations that sound good. The BBB continuously goes after unscrupulous companies that attempt to trick customers into thinking they are members. Nearly every organization has a web page these days; if you suspect something, do a little searching to find out the truth.
Prep before you paint. Make sure to stock up on all the paint supplies you'll need to finish your project. Get enough tarps to cover all your furniture and electronics. Acrylic paints are water based, so they clean up easily and dry fast. Duct tape always comes in handy. Create your own look by mixing two sheens to stripe the wall with paint, or use large format stencils to create a pattern. We can do your paint project for you or do it yourself. If you need a paint sprayer, you have the option to but also consider renting one from us. House Painting Home Painters Highlands Ranch Colorado Click Here
At Custom Design Coatings LLC we offer a variety of painting services applicable for both residential and commercial properties in Emmetsburg, IA. From epoxy flooring to fabulous faux finishes, our Emmetsburg painting contractors can help you cost-effectively transform both the interior and exterior of your property with outstanding results. Our Emmetsburg painting contractors bring a combined 70 years of experience to each and every project, and our expertise for interior and exterior painting is only matched by our dedication to outstanding custom service. Whether you have a small interior painting project or wallpaper removal planned or you are a local business with big commercial painting needs, our Emmetsburg painting contractors are always ready to help! If you need service from a reliable Emmetsburg painting contractor, give Custom Design Coatings LLC a call today, and experience expert quality finishes and true customer service from a team of talented painters that look forward to working for our clients in Emmetsburg, IA.
Home Painters Highlands Ranch Colorado 39.5358216 -104.9716488
Here is where this affects you as a consumer. You select a painter with a contract that says 2 coats, $500 down. You give the company the deposit and pick your colors a couple of days before the project starts. The painter goes to the store with your colors and figures out they are deep base. He (or she) not only needs to charge you more for the paint, but he also needs to charge you for a dark gray primer coat. Ninety nine percent of the time that primer coat is going to be really, really expensive since you already gave a deposit.
Over the past decade, Platinum Painting has grown alongside the community of Arlington and the greater DFW Metroplex. Our team members live right here in the neighborhood with our clients – in fact, we regularly see our work while driving through town! Our professional painters will always treat your home like it's their very own. From the initial complimentary estimate to the final inspection, we'll be working hard to leave you smiling. Give us a chance – you'll be amazed by what we can do!
The PPG logo is a registered trademark and Colorful Communities and We protect and beautify the world are trademarks of PPG Industries Ohio, Inc. © 2019 PPG Industries, Inc. All Rights Reserved. | Legal Notices & Privacy Policies | CA Transparency in Supply Chain Disclosure | Global Code of Ethics | TISC for PPG Architectural Coatings UK Limited | TISC for PPG Industries (UK) Limited | PPG Industries UK Ltd 2017 Gender Pay Gap Report
WOW! I think the guy I hired read this first and I have photos that would make your skin crawl. Bottom line: he got me for $1900.00. Every single thing he painted had to be completely redone....that's when I discovered he did NOT use the colors I picked, he actually used leftover exterior paint from his mother's house! Because I have pets he said things needed to be sealed first and I did agree to that. What I did NOT agree to was using some kind of foul smelling gray stuff ON MY HARDWOOD FLOORS! THEN he painted them BLACK, telling me that all they were good for was covering over with laminate or carpet. He also dripped and tracked paint all over my ceramic tile floors. PLUS left a wet used paint roller in my garden window and had stuff piled in front so I didn't find it until it had dried. I have no idea how much that is going to cost to repair. Then he left without finishing (thank God) but left the "leftover" paint, uncovered, in the rain. Again, hid it so I didn't immediately find it. Obviously we will be going to court but I doubt if I see a penny from him. House Painting Home Painters Highlands Ranch Colorado Click Here
Of the two, interior painting projects are easily the better option for do-it-yourselfers. Since it is not subjected to the elements (or the eyes of passersby), there is a lot more leeway. There are several ways to make DIY interior painting more successful and get closer to professional results, including researching your paint choice, proper wall preparation, understanding good taping techniques and purchasing or renting a professional grade paint sprayer. Before you go it alone, get estimates from a few different contractors and compare them to the hours and material expenses of doing it yourself. While eliminating labor costs is bound to bring some financial advantage, when you add up the materials and the free time spent, you may be surprised at just how reasonable some professional quotes can be.
Lap marks are those ugly stripes caused by uneven layers of paint buildup. They occur when you roll over paint that's already partly dry. (In warm, dry conditions, latex paint can begin to stiffen in less than a minute!) The key to avoiding lap marks when doing DIY wall painting is to maintain a "wet edge," so each stroke of your roller overlaps the previous stroke before the paint can begin to dry.
House Painting Highlands Ranch Colorado 39.5358216 -104.9716488
I was a painting contractor for many years and never asked for a deposit upfront. A good contractor should have great credit with his suppliers and an even better contract if the deal should go south before the job is done. Get to know your customer and explain how each paint covers so they understand fully before you have to apply 3 or 4 coats. If you didn't bid your job right and the client has already signed the contract then that is on you not the client. Yes I haven had to eat the difference but the client was happy and a return customer for years so I made up for it.
Dave, you said it best! Every pro painting contractor truly worth their salt would and should cut and paste exactly what you say here about where customary and legitimate practices and expectations should be in regards to what customers should expect from contractors and how contractors should professionally deal with their customers. By the way, Dave, if you work in the Atlanta area, I would like to hire you! Thank you for your valuable advice!
Over 60 years ago, Graco invented the first portable airless paint sprayer and we've been innovating ever since to deliver the high performance and maximum productivity you need to grow your business. Whether you're a contractor focused on small residential or a large contractor tackling the biggest projects every day, we have a configuration that's designed to meet your needs. Graco airless sprayers are designed and built in the United States of America using the highest-quality parts and backed by the industry's leading warranty. All Graco sprayers are 100% factory-tested and field-proven so you can be sure your money is going toward the best professional paint sprayer you can purchase.
As you walk through your lighted rooms (preferably day light) see if the new coat has light spots showing the precious paint. This is call "bleeding through". This means that there's only one coat of paint or the paint was diluted or the trasition of colors were from light to dark (or the other way around) and primer was not use or the painter is inexperienced.
To paint a large section without leaving lap marks, roll the nearly dry roller in different directions along the dry edge, feathering out the paint as you go. After completing the entire length of the wall or ceiling, move to the next section and paint over the feathered edges. For the second coat, apply the paint in the opposite direction. This crisscrossing paint application sharply reduces (if not eliminates) lap marks. House Painting Home Painters Highlands Ranch Colorado Click Here


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Search over 1,000,000 properties worldwide and 550 airlines worldwide.
No Expedia cancellation fee to change or cancel almost any hotel reservation.
Search over a million flights, hotels, packages, and more
Reviewed on Sep 8, 2019
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Just like a cocktail, your perfect holiday must have just the right mixture. If you are looking for the ideal base for a wonderful trip then search no further than Samarinda. A great location is certainly a must for any good trip, and fortunately for you Samarinda marks off many of the essentials. The next step is to add the finishing garnishes to a vacation you'll want a second round of.
If you want a spot that is the hot spot of the city, then you have found the right place. Some of the most classic mixed drinks like the Martini and Piña Colada were created at a hotel bar in places like Samarinda. No matter how inviting the clubs at these hotels are, you can't sleep in them. Luckily, a stay in Samarinda bar hotels usually offers:
With a hotel that's so lovely from bar booth to bedroom, you might never need to depart.
Your vacation will be off to a wonderful start when you reserve a stay at a hotel with a bar. No matter how your evening unfolds, you know that you'll always have something entertaining to check out in your hotel. Using Expedia, you can get affordable deals on first-class lodging in Samarinda to make your holiday one you happily recall for years to come.


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最后,本文在结构设计的基础上,模拟分析了射流冷却机构在不同射流速度、射流间距、射流口宽的对冷区流场、温度场以及硅熔体上表面平均换热系数的影响。研究结果表明:射流速度越快,射流间距越小,射流口宽度越大,凝固界面处的换热效率越高,带硅的结晶速度就可以越快。根据模拟结果,优化后的射流机构设计参数为射流口宽度为2 mm,射流间距为8 mm。考虑到实验时对温度场以及流场的控制难度问题,选择未来实验时初始射流流速为5 m/s。
At present, the main cost factor thatrestricts the development and utilization of solar energy is the productioncost of silicon wafers. Due to the limitations of current wire-cuttingtechnology, the silicon material loss during the production of silicon wafersis as high as 60%. If the direct drawing of silicon wafers can be realized andthe cutting process is eliminated, the development and utilization costs ofsolar energy will be greatly reduced. To this end, researchers have developed avariety of silicon strip direct-drawing molding manufacturing methods, such asthe fixed-edge film feeding method, the wire traction belt silicon growthmethod, the substrate with silicon growth method and the horizontal stripsilicon growth method. Although the research on the direct stripping technologyof ribbon silicon has been carried out for several decades in the world, theresearch and development of the strip silicon straight-drawing forming processin China is very few at present. In this paper, based on the in-depthunderstanding of foreign studies, a thermal field structure of a horizontalstrip silicon growth furnace was designed, and the structure was optimizedthrough numerical simulation.
In this paper, the theoretical basis andthe influencing mechanism of the thermal field of a horizontal silicon growth furnaceare briefly introduced. Secondly, the melting point and melting rate ofpolysilicon silicon material were measured, and then Alison S. Greenleeproposed the horizontal growth jet mechanism test equipment. The principle ofsilicon wafer growth by floatation silicon put forward by Oliveros GA and thestudy of the line pulling silicon growth furnace were used for reference. Themain structure of the thermal field of the horizontal belt-shaped silicongrowth furnace such as crucibles, heaters, heat insulation layers, and jetstructures were selected and structurally designed. The COMSOL software wasused to simulate the thermal field structure of the horizontal strip silicongrowth furnace, and the rationality of the thermal field structure design of thehorizontal strip silicon growth furnace was demonstrated.
Finally, based on the structural design,this paper simulates and analyzes the effects of jet cooling mechanism on theflow field, temperature field, and the average heat transfer coefficient of theupper surface of the silicon melt at different jet velocity, jet spacing, andjet width. The results show that the faster the jet velocity is, the smallerthe jet spacing is, and the wider the jet port width is, the higher the heattransfer efficiency at the solidification interface is, and the faster thesilicon crystallization speed can be. According to the simulation results, thedesign parameters of the optimized jet mechanism are jet orifice width of 2 mmand jet spacing of 8 mm. Considering the difficulty of controlling thetemperature field and flow field during the experiment, the initial jet flowvelocity is selected to be 5 m/s in the future experiment.
direct forming ; horizontal strip silicongrowth furnace; thermal field; structural design; numerical simulation


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I do not mean to come off with an edge, however, this is an article for people hiring low bid, uninsured, one man band painters out to "hustle" a buck. Hire reputable, established companies or hire someone you know or have a strong referral from. If you are arguing over watered down paint, ticky, tacky deposits, buying your own supplies, you have hired the wrong guy.
Lap marks are those ugly stripes caused by uneven layers of paint buildup. They occur when you roll over paint that's already partly dry. (In warm, dry conditions, latex paint can begin to stiffen in less than a minute!) The key to avoiding lap marks when doing DIY wall painting is to maintain a "wet edge," so each stroke of your roller overlaps the previous stroke before the paint can begin to dry.
It is one of our core values to ensure our clients' dreams for their home comes true. So often, clients come to us for a new coat of paint or a repair but what they really want is so much more. We understand your house is your home and that when it is time for change or for a repair, you want a crew that will take care to make your home look its best.
I would not recommend this company for the following reasons:1. Original appointment was very specific to them on what was needed and to take photos for 3 apartment unit plumbing fixture repairs. One unit had a bathroom tub/shower backup.2. No photos taken, 3 trips later and only 1 unit completed to satisfaction. (Backup) 3. Other faucets unable to find on line with their suppliers and repair 4. Had to find another plumbing company to complete the jobs. 5. Invoicing had to be redone as not in detail of job address for tax purposes 6. Bad communication throughout the project process with their internal customer service. Their technician and project manager ordering parts. 7. For the back up. Did not bring out correct snake on both trips and had to replace lines as too small to break through backup. Costing over $800 for 3 trips just to complete that situation out. 8. Ordered the wrong parts when 2nd trip out since no photos taken to confirm what was needed. Very dissatisfied and will never ask them to work in any of our rental properties again.
Whether you're looking for exterior house painting, interior painting, deck staining, wallpaper removal, wallpapering, wall coverings, roof cleaning, pressure washing or even gutter cleaning, Franklin Painting crews will meet your needs. Our professional house painters are our own employees, not subcontractors. That means when we quote the job, we DO the job. Expect highly trained, professional and courteous employees to make sure your job gets done the way you deserve. Our interior & exterior house painting company is based in Farmington and we cover all of central Connecticut with great painting work.
Professional painters have done this before. It's how they make their living. Through experience, they have learned tricks of the trade and the techniques used to get quality results. They are familiar with the type of paint to use in each situation, know what quality paint is right for each task and know how to achieve a perfectly straight edge – with or without the use of painter's tape.
Texas Professional Painting understands that your home, business, or commercial property is important and that a sound painting job protects your investment. Our skilled San Antonio exterior painters are highly experienced and can tackle any project – whether it is stucco, wood, vinyl, or brick. With Texas Professional Painting, you can get peace of mind that every project will be completed promptly and professionally. Your satisfaction guaranteed.


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text: Martin v. Loewis wrote: > Martijn Faassen <email@example.com> writes: > > > > I think this approach (for determining document order) really does > > > need the notion of identity, so it's hard to believe that this could > > > actually work. > > > > Doesn't __hash__ imply there is a notion of identity? > > No, it implies equality. Actually, it is the other way 'round: > equality implies equal hash values: > > ??? x,y ??? Python-Object: x = y ??? hash(x) = hash(y) This seems to have gotten somewhat mangled here; I get three question marks and this must've been some symbol? Anyway, perhaps the notion of equality is what we need; in my mind two objects can stand in for the same DOM node but not be the same object; they're equal but not identical. The notion for equality in DOM nodes is actually supported by the DOM level 3 working draft: """ isSameNode (introduced in DOM Level 3) Returns whether this node is the same node as the given one. This method provides a way to determine whether two Node references returned by the implementation reference the same object. When two Node references are references to the same object, even if through a proxy, the references may be used completely interchangeably, such that all attributes have the same values and calling the same DOM method on either reference always has exactly the same effect. Issue isSameNode-1: Do we really want to make this different from equals? Resolution: Yes, change name from isIdentical to isSameNode. (Telcon 4 Jul 2000). Issue isSameNode-2: Is this really needed if we provide a unique key? Resolution: Yes, because the key is only unique within a document. (F2F 2 Mar 2001). Issue isSameNode-3: Definition of 'sameness' is needed. The issues are especially interesting, as they're dealing with the same issues we're dealing with now. If I read the text correctly it also implies they are thinking of a notion of a unique 'key', similar to the current id(). This is only implied by some comments in the draft though, and not actually specified anywhere. Then again, I just found out they have a compareTreePosition() method added to the Node interface that we could use for sorting purposes, I think.. > > If a class does not define a __cmp__() method it should not define a > > __hash__() operation either; > > > > I don't understand why this must be so. > > That is surprising indeed; I had expected that this it is the other > way 'round: If you implement __cmp__, you also need to implement > __hash__. It does say that too. :) > > The point is simply that id() is not customizable, and hash() is. > > The point is that hash() is tied to cmp() through the dictionary > implementation, and that conceptionally, hash() maps the large set of > objects to the much smaller set of numbers, in a unidirectional way > (i.e. different objects may have equal hash values). Typically, the > hash of an objects is computed from its state. But that is in fact what is needed in this case; I have many different proxy objects which may all map to the same actual DOM node, so they'd have the same __hash__. But perhaps the other implications of __hash__ break that. What about supplying a 'key' attribute, anticipating DOM level 3 vague implications? :) > > > > * we need to alter the code in xpath/Util.py so as not to rely on > > > > id() anymore but to use the node objects as hash keys directly. > > > > > > Then doing what? What happens if you have two nodes that have the same > > > hash value, but which are different? > > > > I reconsidered later; I simply want to use hash() instead of id. In that > > case you don't have a problem with two objects having the same hash value > > but are different. If they have the same hash value they're identical. > > Not at all. Consider > > <outer> > <inner/> > <inner/> > </outer> > > The two inner elements may reasonably have the same hash value. They > must not be considered identical, since one is before the other in > document order. I don't think it's reasonable to give those inner nodes the same hash value at all. They're not the same node, and shouldn't hash the same way. I don't see any reason to make two different nodes hash the same way just because they have the same name. I'd like to make two node objects hash the same way if they represent the same node in the underlying XML structure. Then again, perhaps we should just move ahead on compareTreePosition or something.. Regards, Martijn


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Off the Cuff With WD-40 Co.'s VP of IT
AGE: 47 (the average age at WD-40)
TITLE: Vice President, Information Technology
COMPANY: WD-40 Co., San Diego
ANNUAL REVENUE: $242 million
INDUSTRY SECTOR: Consumer products
TIME IN JOB: 9.5 years
BEST ADVICE EVER RECEIVED: Care about your customers, but don't worry about them.
BEST TECHNOLOGY DECISION: Consolidating our voice and data with a single vendor
BEST PERSONALITY TRAITS: I like to think I'm a good listener.
WORST PERSONALITY TRAITS: Jumping to conclusions
BIGGEST CAREER INFLUENCE: I've made my career based on people I've worked for and taught myself how not to be, at least in terms of management.
FAVORITE HOBBY WHEN NOT WORKING: Restoring antique vehicles. I drive a 1950s station wagon.
FAVORITE GUILTY PLEASURE: Searching for car parks on eBay. All these old cars -- they break all the time.
HOLIDAY WISH LIST FOR WORK: Time off for the IT staff during the annual Christmas shutdown.
HOLIDAY WISH LIST FOR HOME: A portable DVD player (the "argument eliminator").
FAVORITE QUOTE: "Success is not the result of spontaneous combustion. You must set yourself on fire." -- Fred Shero
Joyce Chutchian was previously managing editor, online for CIO Decisions Media Group. To comment on this story, email email@example.com.


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River Pub
Time Out says
Beautifully upholstered furniture sits on an old tiled floor, while a sturdy bar counter holds up taps of Bass, Caffrey's, Guinness and Kilkenny. The framed photographs from around Istria are a nice touch, old regional maps too, with a few busts of Irish writers. Set behind a big, wooden door, halfway down from the museums.