Revisiting the Auction Algorithm for Weighted Bipartite Perfect Matchings

TL;DR

This paper provides a new, simpler analysis of the auction algorithm for weighted bipartite perfect matchings, leading to tighter bounds on its runtime for approximate solutions in sparse graphs.

Contribution

It introduces a novel analysis technique for the auction algorithm, improving runtime bounds for approximate matchings in regular bipartite graphs.

Findings

Tighter runtime bounds for approximate matchings.

Simpler analysis of the auction algorithm.

Improved performance guarantees in sparse graphs.

Abstract

We study the classical weighted perfect matchings problem for bipartite graphs or sometimes referred to as the assignment problem, i.e., given a weighted bipartite graph $G = (U\cup V,E)$ with weights $w : E \rightarrow \mathcal{R}$ we are interested to find the maximum matching in $G$ with the minimum/maximum weight. In this work we present a new and arguably simpler analysis of one of the earliest techniques developed for solving the assignment problem, namely the auction algorithm. Using our analysis technique we present tighter and improved bounds on the runtime complexity for finding an approximate minumum weight perfect matching in $k$-left regular sparse bipartite graphs.