Multiplicative Auction Algorithm for Approximate Maximum Weight Bipartite Matching

Abstract

We present an auction algorithm using multiplicative instead of constant weight updates to compute a (1 − ε)-approximate maximum weight matching (MWM) in a bipartite graph with n vertices and m edges in time O(mε −1 log(ε −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 − ε)-approximate maximum weight matching under (1) edge deletions in amortized O(ε −1 log(ε −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(ε −1 log(ε −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 − ε)-approximate algorithm in weighted graphs took time O( √ mε −1 log(w max )) per updated edge, where the edge weights lie in the range [1, w max ].