Approximation Algorithms for Minimum-Load <i>k</i> -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 (ML k FL) 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 ⊆ F of k facilities and assign each client j to a facility f ( j )∈ F so as to minimize max f ∈ F ∑ j ∈ C : f ( j )= f d ( f , j ); we call ∑ j ∈ 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 ML k FL for when F = C . ML k FL is rather poorly understood, and only an O ( k )-approximation is currently known for ML k FL, even for line metrics . Our main result is the first polytime approximation scheme (PTAS) for ML k FL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that ML k FL is strongly NP -hard on line metrics. We also devise a quasi-PTAS for ML k FL on tree metrics. ML k FL 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 ML k FL. 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 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 such a structured solution via dynamic programming.

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