09.04.2024 16:00 Lehilton Lelis Chaves Pedrosa:
A parameterized approximation algorithm for the Multiple Allocation k-Hub CenterMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

In the Multiple Allocation k-Hub Center (MAkHC), 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 ⊆ C 2 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 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 we introduce new ideas so that our algorithm approximates the length using only local information.