The red-blue-yellow matching problem

TL;DR

This paper introduces a polynomial-time deterministic algorithm for the red-blue-yellow matching problem, approximating the optimal matching with specific color constraints and leveraging linear programming and topological properties.

Contribution

It extends the red-blue matching problem to three colors, providing the first deterministic approximation algorithm with near-optimal size and exact color counts under natural assumptions.

Findings

Algorithm finds matchings within 3 of optimal size.

Provides a polynomial-time deterministic approach.

Proves a new topological property of plane curves.

Abstract

We consider the red-blue-yellow matching problem: given two natural numbers $k_R$, $k_B$ and a graph $G$ whose edges are colored red, blue or yellow, the goal is to find a matching of $G$ that contains exactly $k_R$ red edges and exactly $k_B$ blue edges, and is of maximum cardinality subject to these constraints. This is a natural generalization of the well known red-blue matching problem, whose complexity status is unknown: although a randomized polynomial-time algorithm exists, a deterministic algorithm has remained elusive for nearly four decades. The best known deterministic approach to the red-blue matching problem, due to Yuster (2012), gives an additive approximation. In this paper, we show a similar result for the red-blue-yellow matching problem, giving a polynomial-time deterministic algorithm that, under natural assumptions, finds a matching satisfying the color requirements almost exactly and has cardinality within 3 of the optimal solution. Our algorithm is a mix of classic linear programming techniques and ad hoc existence results on restricted classes of graphs such as paths and cycles. As a key ingredient, we prove a curious topological property of plane curves, which is a strengthened version of a result by Grandoni and Zenklusen (2010) in the related context of budgeted matchings.