A new algorithm for decremental single-source shortest paths with applications to vertex-capacitated flow and cut problems

We study the vertex-decremental Single-Source Shortest Paths (SSSP) problem: given an undirected graph G=(V,E) with lengths ℓ(e)≥ 1 on its edges that undergoes vertex deletions, and a source vertex s, we need to support (approximate) shortest-path queries in G: given a vertex v, return a path connecting s to v, whose length is at most (1+є) times the length of the shortest such path, where є is a given accuracy parameter. The problem has many applications, for example to flow and cut problems in vertex-capacitated graphs. Decremental SSSP is a fundamental problem in dynamic algorithms that has been studied extensively, especially in the more standard edge-decremental setting, where the input graph G undergoes edge deletions. The classical algorithm of Even and Shiloach supports exact shortest-path queries in O(mn) total update time. A series of recent results have improved this bound to O(m^1+o(1)logL), where L is the largest length of any edge. However, these improved results are randomized algorithms that assume an oblivious adversary. To go beyond the oblivious adversary restriction, recently, Bernstein, and Bernstein and Chechik designed deterministic algorithms for the problem, with total update time Õ(n²logL), that by definition work against an adaptive adversary. Unfortunately, their algorithms introduce a new limitation, namely, they can only return the approximate length of a shortest path, and not the path itself. Many applications of the decremental SSSP problem, including the ones considered in this paper, crucially require both that the algorithm returns the approximate shortest paths themselves and not just their lengths, and that it works against an adaptive adversary. Our main result is a randomized algorithm for vertex-decremental SSSP with total expected update time O(n^2+o(1)logL), that responds to each shortest-path query in Õ(nlogL) time in expectation, returning a (1+є)-approximate shortest path. The algorithm works against an adaptive adversary. The main technical ingredient of our algorithm is an Õ(|E(G)|+ n^1+o(1))-time algorithm to compute a core decomposition of a given dense graph G, which allows us to compute short paths between pairs of query vertices in G efficiently. We use our result for vertex-decremental SSSP to obtain (1+є)-approximation algorithms for maximum s-t flow and minimum s-t cut in vertex-capacitated graphs, in expected time n^2+o(1), and an O(log⁴n)-approximation algorithm for the vertex version of the sparsest cut problem with expected running time n^2+o(1). These results improve upon the previous best known algorithms for these problems in the regime where m= ω(n^{1.5 + o(1)}).