Approximation Algorithms for Minimum-Load k-Facility Location.

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimnm-load klacility location (MLkFL) problem, Which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k >= 0. Assigning client j to a facility f incurs a connection cost d(f, j). The goal is to open a set F subset of F of k facilities and assign each client j to a-facility f (j) is an element of F so as to minimize max(f is an element of F) Sigma(j is an element of C:f(j)=f) d(f, j); we call Sigma(j is an element of C:f(j)=f) d(f, j) the load of facility f. This problem was studied under the name of min-max star cover in References [3, 7], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, a rid only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polytime approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics and resilient to attack by a variety of techniques that have been successfully applied to facility-location problems. For instance, we show that (a) even a configuration-style LP-relaxation has a bad integrality gap and (b) a multi-swap k-median style local search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near optimal solution possessing sonic nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding of the problem and argue that a MILP-solution minimizing a certain potential function possesses the desired structure and then use a rounding algorithm for the generalized assignment problem to "transfer" this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find suds a structured solution via dynamic programming.