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We present dynamic algorithms withpolylogarithmicupdate time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratiostrictly better than 2. Specifically, we obtain a\(1+\frac{1}{\sqrt {2}}+\epsilon \approx 1.707+\epsilon \)approximation in bipartite graphs and a 1.973 + ϵ approximation in general graphs. We thus answer in the affirmative the value version of the major open question repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms’ approximation and worst-case update time bounds both hold w.h.p. against adaptive adversaries.

Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS’21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier.

 

Given an undirected weighted graph with n vertices and m edges, we give the first deterministic m1+o(1)-time algorithm for constructing the cactus representation of all global minimum cuts. This improves the current n2+o(1)-time state-of-the-art deterministic algorithm, which can be obtained by combining ideas implicitly from three papers [22, 27, 12]. The known explicitly stated deterministic algorithm has a runtime of Õ(mn) [9, 34]. Using our technique, we can even speed up the fastest randomized algorithm of [23] whose running time is at least Ω(m log4 n) to O(m log3 n). 

We show the first near-linear time randomized algorithms for listing all minimum vertex cuts of polylogarithmic size that separate the graph into at least three connected components (also known as shredders) and for finding the most shattering one, i.e., the one maximizing the number of connected components. Our algorithms break the quadratic time bound by Cheriyan and Thurimella (STOC'96) for both problems that has been unimproved for more than two decades. Our work also removes an important bottleneck to near-linear time algorithms for the vertex connectivity augmentation problem (Jordan '95) and finding an even-length directed cycle in a graph, a problem shown to be equivalent to many other fundamental problems (Vazirani and Yannakakis '90, Robertson et al. '99). Note that it is necessary to list only minimum vertex cuts that separate the graph into at least three components because there can be an exponential number of minimum vertex cuts in general. To obtain a near-linear time algorithm, we have extended techniques in local flow algorithms developed by Forster et al. (SODA'20) to list shredders on a local scale. We also exploit fast queries to a pairwise vertex connectivity oracle subject to vertex failures (Long and Saranurak FOCS'22, Kosinas ESA'23). This is the first application of using connectivity oracles subject to vertex failures to speed up a static graph algorithm.