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We introduce a new problem that combines the well known All Pairs ShortestPaths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem tocompute the shortest paths for all pairs of vertices for all possible flowamounts. We call this new problem the All Pairs Shortest Paths for All Flows(APSP-AF) problem. We firstly solve the APSP-AF problem on directed graphs withunit edge costs and real edge capacities in$\tilde{O}(\sqrt{t}n^{(\omega+9)/4}) = \tilde{O}(\sqrt{t}n^{2.843})$ time,where $n$ is the number of vertices, $t$ is the number of distinct edgecapacities (flow amounts) and $O(n^{\omega}) < O(n^{2.373})$ is the time takento multiply two $n$-by-$n$ matrices over a ring. Secondly we extend the problemto graphs with positive integer edge costs and present an algorithm with$\tilde{O}(\sqrt{t}c^{(\omega+5)/4}n^{(\omega+9)/4}) =\tilde{O}(\sqrt{t}c^{1.843}n^{2.843})$ worst case time complexity, where $c$ isthe upper bound on edge costs.

Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems