Deterministic near-linear time minimum cut in weighted graphs

In 1996, Karger [Kar96] gave a startling randomized algorithm that finds a minimum-cut in a (weighted) graph in time O(m log3 n) which he termed near-linear time meaning linear (in the size of the input) times a polylogarthmic factor. In this paper, we give the first deterministic algorithm which runs in near-linear time for weighted graphs. Previously, the breakthrough results of Kawarabayashi and Thorup [KT19] gave a near-linear time algorithm for simple graphs (which was improved to have running time O(m log2 n log log n) in [HRW20].) The main technique here is a clustering procedure that perfectly preserves minimum cuts. Recently, Li [Li21] gave an m1+o(1) deterministic minimum-cut algorithm for weighted graphs; this form of running time has been termed “almost-linear”. Li uses almost-linear time deterministic expander decompositions which do not perfectly preserve minimum cuts, but he can use these clusterings to, in a sense, “derandomize” the methods of Karger. In terms of techniques, we provide a structural theorem that says there exists a sparse clustering that preserves minimum cuts in a weighted graph with o(1) error. In addition, we construct it deterministically in near linear time. This was done exactly for simple graphs in [KT19, HRW20] and with polylogarithmic error for weighted graphs in [Li21]. Extending the techniques in [KT19, HRW20] to weighted graphs presents significant challenges, and moreover, the algorithm can only polylogarithmically approximately preserve minimum cuts. A remaining challenge is to reduce the polylogarithmic-approximate clusterings to 1 + o(1/ log n)-approximate so that they can be applied recursively as in [Li21] over O(log n) many levels. This is an additional challenge that requires building on properties of tree-packings in the presence of a wide range of edge weights to, for example, find sources for local flow computations which identify minimum cuts that cross clusters.