Beating Matrix Multiplication for n^1/3-Directed Shortcuts

For an n-vertex digraph G = (V, E) and integer parameter D, a D-shortcut is a small set H of directed edges taken from the transitive closure of G, satisfying that the diameter of G ∪ H is at most D. A recent work [Kogan and Parter, SODA 2022] presented shortcutting algorithms with improved diameter vs. size tradeoffs. Most notably, obtaining linear size D-shortcuts for D = O^e(n^1/3), breaking the √n-diameter barrier. These algorithms run in O(n^ω) time, as they are based on the computation of the transitive closure of the graph. We present a new algorithmic approach for D-shortcuts, that matches the bounds of [Kogan and Parter, SODA 2022], while running in o(n^ω) time for every D ≥ n^1/3. Our approach is based on a reduction to the min-cost max-flow problem, which can be solved in O^e(m + n^3/2) time due to the recent breakthrough result of [Brand et al., STOC 2021]. We also demonstrate the applicability of our techniques to computing the minimal chain covers and dipath decompositions for directed acyclic graphs. For an n-vertex m-edge digraph G = (V, E), our key results are: An Õ(n^1/3 · m + n^3/2)-time algorithm for computing D-shortcuts of linear size for D = Õ(n^1/3), and an Õ(n^1/4 · m + n^7/4)-time algorithm for computing D-shortcuts of Õ(n^3/4) edges for D = Õ(n^1/2). For a DAG G, we provide Õ(m + n^3/2)-time algorithms for computing its minimum chain covers, maximum antichain, and decomposition into dipaths and independent sets. This improves considerably over the state-of-the-art bounds by [Caceres et al., SODA 2022] and [Grandoni et al., SODA 2021]. Our results also provide a new connection between shortcutting sets and the seemingly less related problems of minimum chain covers and the maximum antichains in DAGs.