A DFS Algorithm for Maximum Matchings in General Graphs

TL;DR

This paper introduces a DFS-based algorithm for finding maximum matchings in general graphs that avoids blossom shrinking, using detour information to improve efficiency and operate in O(mn) time.

Contribution

It presents a novel DFS algorithm that bypasses blossom shrinking, reducing complexity and maintaining efficiency in maximum matching computations.

Findings

Operates in O(mn) time for maximum matchings

Uses auxiliary stack to store detour information

Avoids blossom shrinking, simplifying implementation

Abstract

In this paper, we propose a depth-first search (DFS) algorithm for searching maximum matchings in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating paths in the super-vertices shrunk from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with $m$ edges and $n$ vertices in $O(mn)$ time with space complexity $O(n)$.