Multiplicative Auction Algorithm for Approximate Maximum Weight Bipartite Matching

TL;DR

This paper introduces a simple multiplicative auction algorithm for approximate maximum weight bipartite matching that outperforms previous algorithms in runtime and can be extended to dynamic graph updates.

Contribution

The paper presents a novel multiplicative update auction algorithm achieving faster approximation of maximum weight matching and extends it to dynamic graph scenarios.

Findings

Achieves $O(m ext{ extbackslash}eps^{-1})$ runtime for $(1- ext{ extbackslash}eps)$-approximate MWM.

Outperforms the previous best $O(m ext{ extbackslash}eps^{-1} ext{ extbackslash}log ext{ extbackslash}eps^{-1})$ algorithm.

Can be extended to maintain approximate MWM under dynamic vertex insertions and deletions.

Abstract

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