Efficient parameterized algorithms for computing all-pairs shortest paths

Abstract

Computing for all pairs of vertices the shortest paths in a graph is a fundamental and much-studied problem with many applications. Unfortunately, despite intense study, there are still no significantly faster algorithms for it than the O(n3) time algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist for the vertex-weighted version if fast matrix multiplication may be used. Yuster (SODA 2009) gave an algorithm running in time O(n2.842), but no combinatorial, truly subcubic algorithm is known.Motivated by the recent framework of efficient parameterized algorithms (or “FPT in P”), we investigate the influence of the graph parameters clique-width (cw) and modular-width (mw) on the running times of algorithms for solving vertex-weighted all-pairs shortest paths. We obtain efficient (and combinatorial) parameterized algorithms of times O(cw2n2), resp. O(mw2n+n2). If fast matrix multiplication is allowed then the latter can be improved to O(mw1.842n+n2) using the algorithm of Yuster as a black box. The algorithm relative to modular-width is adaptive, meaning that the running time matches the best unparameterized algorithm for parameter value mw equal to n, and outperforms it already for mw∈O(n1−ɛ) for any ɛ>0.