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 $\mathcal{O}(n^3)$ 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 $\mathcal{O}(n^{2.842})$, 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 $\mathcal{O}(cw^2n^2)$, resp. $\mathcal{O}(mw^2n + n^2)$. If fast matrix multiplication is allowed then the latter can be improved to $\mathcal{O}(mw^{1.842}n + n^2)$ 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 $mw \in \mathcal{O}(n^{1 - \varepsilon})$ for any $\varepsilon > 0$.

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用于计算所有对最短路径的高效参数化算法

计算所有对最短路径是许多应用程序中一个基本且经过大量研究的问题。不幸的是，尽管进行了大量研究，但由于 Floyd 和 Warshall（1962），仍然没有比 $\mathcal{O}(n^3)$ 时间算法更快的算法。如果可以使用快速矩阵乘法，则顶点加权版本存在更快的算法。Yuster (SODA 2009) 给出了一个在时间 $\mathcal{O}(n^{2.842})$ 中运行的算法，但没有已知的组合，真正的亚三次算法。受最近的高效参数化算法框架（或“P 中的 FPT”）的启发，我们研究了图参数 clique-width ($cw$) 和模块化宽度 ($mw$) 对算法运行时间的影响解决所有对最短路径。我们在时间 $\mathcal{O}(cw^2n^2)$ 的非负顶点加权图上获得了有效的（和组合的）参数化算法。$\mathcal{O}(mw^2n + n^2)$。如果允许快速矩阵乘法，则可以使用 Yuster 算法作为黑盒将后者改进为 $\mathcal{O}(mw^{1.842}n + n^2)$。相对于模块化宽度的算法是自适应的，这意味着运行时间匹配参数值 $mw$ 等于 $n$ 的最佳非参数化算法，并且它们在 $mw \in \mathcal{O}(n^ {1 - \varepsilon})$ 对于任何 $\varepsilon > 0$。