Near approximation of maximum weight matching through efficient weight reduction
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Linear programming in the semi-streaming model with application to the maximum matching problem
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Efficient algorithms for maximum weight matchings in general graphs with small edge weights
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Linear programming in the semi-streaming model with application to the maximum matching problem
Information and Computation
Linear-Time Approximation for Maximum Weight Matching
Journal of the ACM (JACM)
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Given a weighted graph, the {\em maximum weight matching} problem (MWM) is to find a set of vertex-disjoint edges with maximum weight. In the 1960s Edmonds showed that MWMs can be found in polynomial time. At present the fastest MWM algorithm, due to Gabow and Tarjan, runs in $\tilde{O}(m\sqrt{n})$ time, where $m$ and $n$ are the number of edges and vertices in the graph. Surprisingly, restricted versions of the problem, such as computing $(1-\epsilon)$-approximate MWMs or finding maximum cardinality matchings, are not known to be much easier (on sparse graphs). The best algorithms for these problems also run in $\tilde{O}(m\sqrt{n})$ time. In this paper we present the first near-linear time algorithm for computing $(1-\epsilon)$-approximate MWMs. Specifically, given an arbitrary real-weighted graph and $\epsilon0$, our algorithm computes such a matching in $O(m\epsilon^{-2}\log^3 n)$ time. The previous best approximate MWM algorithm with comparable running time could only guarantee a $(2/3-\epsilon)$-approximate solution. In addition, we present a faster algorithm, running in $O(m\log n\log\epsilon^{-1})$ time, that computes a $(3/4-\epsilon)$-approximate MWM.