Faster and Unified Algorithms for Diameter Reducing Shortcuts and Minimum Chain Covers

For an n-vertex m-edge digraph G, 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. In a sequence of works [Kogan and Parter, SODA 2022 ICALP 2022] provided shortcut algorithms with improved diameter vs. size tradeoffs. In this paper, we present faster and unified shortcut algorithms for general digraphs. These algorithms also yield improved tradeoffs for the family of bounded-width DAGs. We show: • A unified and faster shortcutting algorithm which implements the [KP,SODA'22] framework in almost optimal time, conditioned on the combinatorial Boolean Matrix Multiplication (BMM) conjecture. For example, Õ(n1/3)-shortcuts with Õ(n) edges can be computed in time Õ(n1/3 · m); and Õ(n1/2)- shortcuts with Õ(n3/4) edges can be computed in time Õ(n1/4 · m). This improves the time bounds of Õ(n1/3 · m + n3/2) and Õ(n1/4 · m + n7/4) respectively, by [KP, ICALP'22]. • An improved algorithm for computing a Minimum Chain Cover (MCC) of DAGs. For an n-vertex m-edge DAG G of width k, the algorithm runs in Õ(√k · n + m1+o(1) time. For sparse digraphs, we show faster Õ(k1/3 · n + n1+o(1))-time algorithms. This improves the time bounds of Õ(n3/2 + m) [KP, ICALP'22] and Õ(k2 · n + m) [Caceres et al., SODA 2022]. • An MCC-based shortcut algorithm for DAGs with improved size and time bounds, as a function of the width k. For example, providing a linear-size -√k-shortcut in time Õ(min√k · m + m1+o(1),n2), improving the general graph's size and time bounds for k = o(n2/3).