New parameterized algorithms for APSP in directed graphs

All Pairs Shortest Path (APSP) is a classic problem in graph theory. While for general weighted graphs there is no algorithm that computes APSP in O(n) time (ϵ > 0), by using fast matrix multiplication algorithms, we can compute APSP in O(n^ω logn) time (ω < 2.373) for undirected unweighted graphs, and in O(n^2.5302) time for directed unweighted graphs. In the current state of matters, there is a substantial gap between the upper bounds of the problem for undirected and directed graphs, and for a long time, it is remained an important open question whether it is possible to close this gap. In this paper we introduce a new parameter that measures the symmetry of directed graphs (i.e. their closeness to undirected graphs), and obtain a new parameterized APSP algorithm for directed unweighted graphs, that generalizes Seidel's O(n^ω logn) time algorithm for undirected unweighted graphs. Given a directed unweighted graph G, unless it is highly asymmetric, our algorithms can compute APSP in o(n^2.5) time for G, providing for such graphs a faster APSP algorithm than the state-of-the-art algorithms for the problem.