On uniform capacitated a: Median beyond the natural LP relaxation

In this paper, we study the uniform capacitated k-median problem. In the problem, we are given a set J- of potential facility locations, a set C of clients, a metric d over F ∪ C, an upper bound k on the number of facilities we can open and an upper bound u on the number of clients each facility can serve. We need to open a subset S ⊆ T of k facilities and connect clients in C to facilities in S so that each facility is connected by at most u clients. The goal is to minimize the total connection cost over all clients. Obtaining a constant approximation algorithm for this problem is a notorious open problem; most previous works gave constant approximations by either violating the capacity constraints or the cardinality constraint. Notably, all these algorithms are based on the natural LP-relaxation for the problem. The LP-relaxation has unbounded integrality gap, even when we are allowed to violate the capacity constraints or the cardinality constraint by a factor of 2 - ε. Our result is an exp(O(l/ε²))-approximation algorithm for the problem that violates the cardinality constraint by a factor of 1 + ε. That is, we find a solution that opens at most (1 + ε)k facilities whose cost is at most exp(0(l/e2)) times the optimum solution when at most k facilities can be open. This is already beyond the capability of the natural LP relaxation, as it has unbounded integrality gap even if we are allowed to open (2 - ε)k facilities. Indeed, our result is based on a novel LP for this problem. We hope that this LP is the first step towards a constant approximation for capacitated κ-median. The version as we described is the hard-capacitated version of the problem, as we can only open one facility at each location. This is as opposed to the soft-capacitated version, in which we are allowed to open more than one facilities at each location. The hard-capacitated version is more general, since one can convert a soft-capacitated instance to a hard-capacitated instance by making enough copies of each facility location. We give a simple proof that in the uniform capacitated case, the soft-capacitated version and the hard-capacitated version are actually equivalent, up to a small constant loss in the approximation ratio. Moreover, we show that the given potential facility locations do not matter: we can assume T = C.