Approximating capacitated $k$-median with $(1+\epsilon)k$ open facilities

In the capacitated k-median (CKM) problem, we are given a set F of facilities, each with a capacity, a set C of clients, a metric d over F ∪ C and an integer k. The goal is to open k facilities in F and connect the clients C to the open facilities subject to the capacity constraints, so as to minimize the total connection cost. In this paper, we give a constant approximation for CKM, by violating the cardinality constraint by a factor of 1+ . This generalizes the result of [Li15], which only works for the uniform capacitated case. Our algorithm gives the first constant approximation for general CKM that only opens (1 + )k facilities. Indeed, most previous algorithms for CKM are based on the natural LP relaxation for the problem, which has unbounded integrality gap even if (2− )k facilities can be opened. Instead, our algorithm is based on a novel configuration LP for the problem. For each set B ⊆ F of potential facilities, we try to characterize the convex hull of all valid integral solutions projected to the instance defined by B and C. This is of course impossible as this goal for the set B = F is equivalent to solve the CKM problem. As we only care about the bad situation when there are at most a constant number `1 of open facilities in B, we cut the convex hull into two parts: conditioned on the event that at most `1 facilities are open in B we have the exact polytope; conditioned on the event that more than `1 facilities are open, we only have a relaxed polytope. This allows us to reduce the size of the polytope to n1. This LP can not be solved efficiently as there are exponential number of sets B. Instead, we use the standard trick: given a fractional solution, our rounding algorithm either succeeds, or finds a set B such that the fractional solution is not valid for the set B. This allows us to combine the rounding algorithm with the ellipsoid method. ∗TTIC, [email protected] ar X iv :1 41 1. 56 30 v1 [ cs .D S] 2 0 N ov 2 01 4