红-蓝-黄匹配问题
The red-blue-yellow matching problem
AI总结：
研究红-蓝-黄匹配问题，提出一种确定性多项式时间算法，在自然假设下找到几乎满足颜色要求且基数与最优解相差不超过3的匹配。
摘要：
我们考虑红-蓝-黄匹配问题：给定两个自然数 $k_R$, $k_B$ 和一个边被染成红色、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，并且在这些约束下基数最大。这是著名的红-蓝匹配问题的自然推广，其复杂性状态未知：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找到。已知最好的红-蓝匹配问题的确定性方法由 Yuster (2012) 提出，给出了一个加性近似。在本文中，我们展示了红-蓝-黄匹配问题的类似结果，给出了一种确定性多项式时间算法，在自然假设下，该算法几乎精确地找到满足颜色要求的匹配，且基数与最优解相差不超过3。我们的算法是经典线性规划技术和关于受限图类（如路径和环）的特定存在性结果的混合。作为一个关键组成部分，我们证明了平面曲线的一个奇特的拓扑性质，这是 Grandoni 和 Zenklusen (2010) 在预算匹配相关背景下结果的加强版本。
英文摘要：
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.
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