A Parameterized Approximation Algorithm for the Multiple Allocation k-Hub Center

Abstract

In the Multiple Allocation k -Hub Center, we are given a connected edge-weighted graph G, sets of clients C and hub locations H, where V(G) = C∪ H, a set of demands D⊆ C2 and a positive integer k. A solution is a set of hubs H⊆ H of size k such that every demand (a, b) is satisfied by a path starting in a, going through some vertex of H, and ending in b. The objective is to minimize the largest length of a path. We show that finding a (3 - ϵ) -approximation is NP-hard already for planar graphs. For arbitrary graphs, the approximation lower bound holds even if we parameterize by k and the value r of an optimal solution. An exact FPT algorithm is also unlikely when the parameter combines k and various graph widths, including pathwidth. To confront these hardness barriers, we give a (2 + ϵ) -approximation algorithm parameterized by treewidth, and, as a byproduct, for unweighted planar graphs, we give a (2 + ϵ) -approximation algorithm parameterized by k and r. Compared to classical location problems, computing the length of a path depends on non-local decisions. This turns standard dynamic programming algorithms impractical, thus our algorithm approximates this length using only local information.