Multiplicative Auction Algorithm for Approximate Maximum Weight Bipartite Matching

$$We present an auction algorithm using multiplicative instead of constant weight updates to compute a $(1-\text{eps})$-approximate maximum weight matching (MWM) in a bipartite graph with $n$ vertices and $m$ edges in time $O(m{\text{eps}}^{-1})$, beating the running time of the fastest known approximation algorithm of Duan and Pettie [JACM '14] that runs in $O(m{\text{eps}}^{-1}\log {\text{eps}}^{-1})$. Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a $(1-\text{eps})$-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 used is $O(m{\text{eps}}^{-1})$, where $m$ is the sum of the number of initially existing and inserted edges.