Dynamic Algorithms for Maximum Matching Size
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We study fully dynamic algorithms for maximum matching. This is a well-studied problem, known to admit several update-time/approximation trade-offs. For instance, it is known how to maintain a 1/2-approximate matching in (polylog n) time or a 2/3-approximate matching in O(√(n)) time, where n is the number of vertices. Improving either of these bounds has been a long-standing open problem. In this paper, we show that when the goal is to maintain just the size of the matching instead of its edge-set, then these bounds can indeed be improved. We give algorithms that maintain * a .501-approximation in (polylog n) update-time for general graphs, * a .534-approximation in (polylog n) update-time for bipartite graphs, and * a (2/3 + Ω(1))-approximation in O(√(n)) update-time for bipartite graphs.
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