Fully Dynamic Almost-Maximal Matching: Breaking the Polynomial Barrier for Worst-Case Time Bounds
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Despite significant research efforts, the state-of-the-art algorithm for maintaining an approximate matching in fully dynamic graphs has a polynomial worst-case update time, even for very poor approximation guarantees. In a recent breakthrough, Bhattacharya, Henzinger and Nanongkai showed how to maintain a constant approximation to the minimum vertex cover, and thus also a constant-factor estimate of the maximum matching size, with polylogarithmic worst-case update time. Later (in SODA'17 Proc.) they improved the approximation factor all the way to 2+. Nevertheless, the longstanding fundamental problem of maintaining an approximate matching with sub-polynomial worst-case time bounds remained open. In this work we present a randomized algorithm for maintaining an almost-maximal matching in fully dynamic graphs with polylogarithmic worst-case update time. Such a matching provides (2+)-approximations for both the maximum matching and the minimum vertex cover, for any > 0. Our result was done independently of the SODA'17 (2+)-approximation result of Bhattacharya et al., so it provides the first (2+)-approximation for minimum vertex cover (together with Bhattacharya et al.'s result) and the first (2+)-approximation for maximum (integral) matching. The worst-case update time of our algorithm, namely O(( n,^-1)), holds deterministically, while the almost-maximality guarantee holds with high probability. Moreover, this guarantee also bounds the number of changes (replacements) to our matching in the worst-case. This result not only settles the aforementioned problem on dynamic matchings, but also provides essentially the best possible (under the unique games conjecture) approximation guarantee for dynamic vertex cover.
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