The red–blue–yellow matching problem

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.