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Improved approximation algorithms for the minimum latency problem via prize-collecting strolls

The minimum latency problem (MLP) is a well-studied variant of the traveling salesman problem (TSP). In the MLP, the server's goal is to minimize the average latency that the clients experience prior to being served, rather than the total latency experienced by the server (as in the TSP). The MLP sometimes goes by other names, such as the traveling repairman problem, or the deliveryman problem. Unlike most combinatorial optimization problems, the MLP is NP-hard even on trees (Sitters, 2001). Our main result is an improved approximation algorithm for the MLP on trees, upon which we build improved approximation algorithms for a much wider class of graphs.
 The MLP on trees is interesting for several reasons. First, many of the aspects that make the problem difficult on general graphs are already present in the tree case. Second, all existing approximation algorithms for general graphs are built on approximation algorithms for the tree case. Third, there has been no improvement for the tree case since the 3.59-approximation of Goemans and Kleinberg, first introduced 14 years ago in 1996. Fourth, in the intervening period, the best ratio for general metrics has been improved to match the 3.59 for trees (Chaudhuri et al., 2003).
 In this paper, we improve the approximation ratio for trees to 3.03. In fact, our 3.03-approximation algorithm works for any class of graphs in which the related prize-collecting stroll (PCS) problem is solvable in polynomial time, such as graphs of constant treewidth. More generally, for any class of graphs that admit a Lagrangian-preserving β-approximation algorithm, we can use this algorithm as a black box to achieve a 3.03β-approximation for the MLP. Sadly, this does not immediately improve the ratio of 3.59 for general graphs, because the current best value of β for that case is 2.
 One interesting piece of our analysis is the solution of an infinite-dimensional linear program, used to analyze a finite-dimensional factor-revealing linear program (FRLP). We believe that our methods may hold promise for easing the analysis of other FRLPs encountered in the literature.