Efficient parameterized algorithms for computing all-pairs shortest paths
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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 O(n^3) 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(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 O(cw^2n^2), resp. O(mw^2n + n^2). If fast matrix multiplication is allowed then the latter can be improved to O(mw^1.842n + n^2) 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 ∈O(n^1 - ε) for any ε > 0.
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