摘要：Computing all-pairs shortest paths 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 ð'ª(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 ð'ª(n^2.842), 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 ALL-PAIRS SHORTEST PATHS. We obtain efficient (and combinatorial) parameterized algorithms on non-negative vertex-weighted graphs of times ð'ª(cw²n²), resp. ð'ª(mw²n + n²). If fast matrix multiplication is allowed then the latter can be improved to ð'ª(mw^{1.842} n + 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 they outperform them already for mw â^^ ð'ª(n^{1 - ε}) for any ε > 0.
