Near approximation of maximum weight matching through efficient weight reduction

TL;DR

This paper presents a method to efficiently approximate maximum weight matchings in hypergraphs by reducing weights, enabling near-optimal solutions with improved runtime complexity.

Contribution

It introduces a weight reduction technique that, combined with existing algorithms, achieves near-approximate maximum weight matchings in hypergraphs with improved efficiency.

Findings

Achieves (1-ε)-approximation for maximum weight matching in graphs.

Provides a new weight reduction approach for hypergraph matchings.

Improves runtime complexity for approximate maximum weight matching algorithms.

Abstract

Let G be an edge-weighted hypergraph on n vertices, m edges of size \le s, where the edges have real weights in an interval [1,W]. We show that if we can approximate a maximum weight matching in G within factor alpha in time T(n,m,W) then we can find a matching of weight at least (alpha-epsilon) times the maximum weight of a matching in G in time (epsilon^{-1})^{O(1)}max_{1\le q \le O(epsilon \frac {log {\frac n {epsilon}}} {log epsilon^{-1}})} max_{m_1+...m_q=m} sum_1^qT(min{n,sm_j},m_{j},(epsilon^{-1})^{O(epsilon^{-1})}). In particular, if we combine our result with the recent (1-\epsilon)-approximation algorithm for maximum weight matching in graphs due to Duan and Pettie whose time complexity has a poly-logarithmic dependence on W then we obtain a (1-\epsilon)-approximation algorithm for maximum weight matching in graphs running in time (epsilon^{-1})^{O(1)}(m+n).