Multiplicative Auction Algorithm for Approximate Maximum Weight Bipartite Matching
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We present an auction algorithm using multiplicative instead of constant weight updates to compute a (1-)-approximate maximum weight matching (MWM) in a bipartite graph with n vertices and m edges in time O(m^-1log(^-1)), matching the running time of the linear-time approximation algorithm of Duan and Pettie [JACM '14]. Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a (1-)-approximate maximum weight matching under (1) edge deletions in amortized O(^-1log(^-1)) time and (2) one-sided vertex insertions. If all edges incident to an inserted vertex are given in sorted weight the amortized time is O(^-1log(^-1)) per inserted edge. If the inserted incident edges are not sorted, the amortized time per inserted edge increases by an additive term of O(log n). The fastest prior dynamic (1-)-approximate algorithm in weighted graphs took time O(√(m)^-1log (w_max)) per updated edge, where the edge weights lie in the range [1,w_max].
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