K-medians, facility location, and the Chernoff-Wald bound

We study the general (non-metric) facility-location and weighted k-medians problems, as well as the fractional facility-location and k-medians problems. We describe a natural randomized rounding scheme and use it to derive approximation algorithms for all of these problems. For facility location and weighted k-medians, the respective algorithms are polynomial-time [H∆k + d]and [(1 + ǫ)d; ln(n + n/ǫ)k]-approximation algorithms. These performance guarantees improve on the best previous performance guarantees, due respectively to Hochbaum (1982) and Lin and Vitter (1992). For fractional k-medians, the algorithm is a new, Lagrangian-relaxation, [(1+ ǫ)d, (1+ ǫ)k]approximation algorithm. It runs in O(k ln(n/ǫ)/ǫ) lineartime iterations. For fractional facilities-location (a generalization of fractional weighted set cover), the algorithm is a Lagrangianrelaxation, [(1 + ǫ)k]-approximation algorithm. It runs in O(n ln(n)/ǫ) linear-time iterations and is essentially the same as an unpublished Lagrangian-relaxation algorithm due to Garg (1998). By recasting his analysis probabilistically and abstracting it, we obtain an interesting (and as far as we know new) probabilistic bound that may be of independent interest. We call it the Chernoff-Wald bound. 1 Problem definitions The input to the weighted set cover problem is a collection of sets, where each set s is given a cost cost(s) ∈ R+. The goal is to choose a cover (a collection of sets containing all elements) of minimum total cost. The (uncapacitated) facility-location problem is a generalization of weighted set cover in which each set f (called a “facility”) and element c (called a “customer”) are given a distance dist(f, c) ∈ R+∪{∞}. The goal is to choose a set of facilities F minimizing cost(F ) + dist(F ), where cost(F ), the facility cost of F , is ∑ f∈F cost(f) and dist(F ), the assignment cost of F , is ∑ c minf∈F dist(f, c). Fig. 1 shows the standard integer programming formulation of the problem — the facility-location IP [12, p. 8]. The facility-location linear program (LP) is the same except without the constraint “x(f) ∈ {0, 1}”. A fractional solution is a feasible solution to the LP. Fractional facility location is the problem of solving the LP. Research partially funded by NSF CAREER award CCR9720664. Department of Computer Science, Dartmouth College, Hanover, NH. [email protected] minimizex d+ k subject to    cost(x) ≤ k dist(x) ≤ d ∑ f x(f, c) = 1 (∀c) x(f, c) ≤ x(f) (∀f, c)