Improved Approximation Algorithms for the Matroid Median Problem

We consider the matroid median problem (MMP), which is defined as follows. As in the uncapacitated facility location problem, we are given a set of facilities F and a set of clients D. Each facility i has an opening cost of fi. Each client j ∈ D has demand dj and assigning client j to facility i incurs an assignment cost of djcij proportional to the distance between i and j. Further, we are given a matroid M = (F , I) on the set of facilities. The goal is to choose a set F ∈ I of facilities to open that forms an independent set in M , and assign each client j to a facility i(j) ∈ F so as to minimize the total facility-opening and client-assignment costs, that is, ∑ i∈F fi+ ∑ j∈D ci(j)j . We assume that the facilities and clients are located in a common metric space, so the distances cij form a metric. The matroid median problem is a generalization of the metric k-median problem, which is the special case where M is a uniform matroid (there are no facility-opening costs), and is thus, NP-hard. The matroid median problem without facility-opening costs was introduced very recently by Krishnaswamy et al. [4], who gave a 16-approximation algorithm for this problem. We devise an improved 10-approximation algorithm for this problem (Section 3). Moreover, notably, our algorithm is significantly simpler and cleaner than the one in [4]. The effectiveness of our simpler approach for matroid median is further highlighted when we consider the matroid median problem with penalties, which is the generalization of matroid median where we are allowed to not assign a client to an open facility at the expense of incurring a certain penalty for each such unassigned client. We leverage the techniques underlying our simpler and cleaner algorithm for matroid median to devise a 34-approximation algorithm (Section 4), which is a vast improvement over the the approximation ratio of 360 obtained by Krishaswamy et al. [4]. Our improvement comes from an improved and simpler rounding procedure for a natural LP relaxation of the problem also considered in [4]. We show that a clustering step introduced in [1] for the k-median problem coupled with two applications of the integrality of the intersection of two submodular (or matroid) polyhedra—one to obtain a half-integral solution, and another to obtain an integral solution—suffices to obtain the desired approximation ratio. In contrast, the algorithm in [4] starts off with the clustering step in [1], but then further dovetails the rounding procedure of [1] creating trees and then stars and then applies the integrality of the intersection of two submodular polyhedra.