Multiplicative Auction Algorithm for Approximate Maximum Weight Bipartite Matching

Da Wei Zheng, Monika Henzinger

Abstract: We present an auction algorithm using multiplicative instead of constant weight updates to compute a \((1-\varepsilon )\)-approximate maximum weight matching (MWM) in a bipartite graph with n vertices and m edges in time \(O(m\varepsilon ^{-1}\log (\varepsilon ^{-1}))\), 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 \((1-\varepsilon )\)-approximate maximum weight matching under (1) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time time used is \(O(m\varepsilon ^{-1}\log (\varepsilon ^{-1}))\), where m is the sum of the number of initially existing and inserted edges.