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.

翻译：考虑红-蓝-黄匹配问题：给定两个自然数 $k_R$、$k_B$ 以及一个边被染成红色、蓝色或黄色的图 $G$，目标是找到 $G$ 的一个匹配，该匹配恰好包含 $k_R$ 条红边和 $k_B$ 条蓝边，并在这些约束下具有最大基数。这是著名的红-蓝匹配问题的自然推广，而后者的复杂性状态尚不明确：尽管存在随机多项式时间算法，但近四十年来确定性算法一直难以获得。已知解决红-蓝匹配问题的最佳确定性方法由 Yuster (2012) 提出，仅能给出加性近似。在本文中，我们针对红-蓝-黄匹配问题展示了类似结果，提出一个多项式时间的确定性算法，该算法在自然假设下能几乎精确地满足颜色要求，且解的基数与最优解相差不超过 3。我们的算法融合了经典线性规划技术和针对路径、环等受限图类的特设存在性结果。作为关键要素，我们证明了平面曲线的一个奇特拓扑性质，这是 Grandoni 和 Zenklusen (2010) 在预算匹配相关背景下所得结论的强化版本。