Maximal k-Edge-Connected Subgraphs in Almost-Linear Time for Small k
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We give the first almost-linear time algorithm for computing the maximal k-edge-connected subgraphs of an undirected unweighted graph for any constant k. More specifically, given an n-vertex m-edge graph G=(V,E) and a number k = log^o(1)n, we can deterministically compute in O(m+n^1+o(1)) time the unique vertex partition {V_1,…,V_z} such that, for every i, V_i induces a k-edge-connected subgraph while every superset V'_i⊃ V_i does not. Previous algorithms with linear time work only when k≤2 [Tarjan SICOMP'72], otherwise they all require Ω(m+n√(n)) time even when k=3 [Chechik et al. SODA'17; Forster et al. SODA'20]. Our algorithm also extends to the decremental graph setting; we can deterministically maintain the maximal k-edge-connected subgraphs of a graph undergoing edge deletions in m^1+o(1) total update time. Our key idea is a reduction to the dynamic algorithm supporting pairwise k-edge-connectivity queries [Jin and Sun FOCS'20].
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