Near-Optimal Dynamic Rounding of Fractional Matchings in Bipartite Graphs
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We study dynamic (1-ϵ)-approximate rounding of fractional matchings – a key ingredient in numerous breakthroughs in the dynamic graph algorithms literature. Our first contribution is a surprisingly simple deterministic rounding algorithm in bipartite graphs with amortized update time O(ϵ^-1log^2 (ϵ^-1· n)), matching an (unconditional) recourse lower bound of Ω(ϵ^-1) up to logarithmic factors. Moreover, this algorithm's update time improves provided the minimum (non-zero) weight in the fractional matching is lower bounded throughout. Combining this algorithm with novel dynamic partial rounding algorithms to increase this minimum weight, we obtain several algorithms that improve this dependence on n. For example, we give a high-probability randomized algorithm with Õ(ϵ^-1· (loglog n)^2)-update time against adaptive adversaries. (We use Soft-Oh notation, Õ, to suppress polylogarithmic factors in the argument, i.e., Õ(f)=O(f·poly(log f)).) Using our rounding algorithms, we also round known (1-ϵ)-decremental fractional bipartite matching algorithms with no asymptotic overhead, thus improving on state-of-the-art algorithms for the decremental bipartite matching problem. Further, we provide extensions of our results to general graphs and to maintaining almost-maximal matchings.
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