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 minimum-load k-facility 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' of k facilities, and assign each client j to a facility f(j) in F' so as to minimize maximum, over all facilities in F', of the sum of distances of clients j assigned to F' to F'. We call this sum the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polynomial time 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 the 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 some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding 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 such a structured solution via dynamic programming.