Reducing Maximum Weighted Matching to the Largest Cardinality Matching in CONGEST

TL;DR

This paper introduces a new CONGEST algorithm that reduces the maximum weighted matching problem to the largest cardinality matching problem in constant rounds, using a novel rounding technique and assumptions about edge-weight knowledge.

Contribution

The paper presents a simple, efficient CONGEST algorithm for reduction, and a rounding method to handle general instances, advancing distributed matching algorithms.

Findings

Constant-round reduction in CONGEST

Effective rounding for weighted matching

Applicability to general weighted matching instances

Abstract

In this paper, we reduce the maximum weighted matching problem to the largest cardinality matching in {\bf CONGEST}. The paper presents two technical contributions. The first of them is a simple $poly(\log n, \frac{1}{\varepsilon}, t, \ln w_t)$-round {\bf CONGEST} algorithm for reducing the maximum weighted matching problem to the largest cardinality matching problem. This is achieved under the assumption that all vertices know all edge-weights $\{w_1,....,w_t\}$ (in particular, they know $t$, the number of different edge-weights), though a particular vertex may not know the weight of a particular edge. Our second ingredient is a simple rounding algorithm (similar to approximation algorithms for the bin packing problem) allowing to reduce general instances of the maximum weighted matching problem to ones satisfying the assumptions of the first ingredient, in which $t\leq poly'(\log n, \frac{1}{\varepsilon})$. We end the paper with a brief discussion of implementing our algorithms in {\bf CONGEST}. Our main conclusion is that we just need constant rounds for the reduction.