Efficient parameterized algorithms for computing allpairs shortest paths
Abstract
Computing allpairs shortest paths is a fundamental and muchstudied 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 vertexweighted 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 cliquewidth ($cw$) and modularwidth ($mw$) on the running times of algorithms for solving AllPairs Shortest Paths. We obtain efficient (and combinatorial) parameterized algorithms on nonnegative vertexweighted 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 modularwidth 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 $\varepsilon > 0$.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.04908
 Bibcode:
 2020arXiv200104908K
 Keywords:

 Computer Science  Data Structures and Algorithms