最大匹配-匹配：困难性与近似算法

中文摘要 AI 辅助

本文研究了Iburi和Uehara（FUN 2024）提出的匹配-匹配谜题的优化变体\ extsc{MaxMMP}。给定一个图、一个部分顶点着色以及一个彩色棍棒的多重集，目标是完成着色并将棍棒分配给图的边，以最大化满足条件的边的数量。我们首先通过从\ extsc{Max-Cut}归约证明\ extsc{MaxMMP}是APX-hard的。该困难性在仅使用两种颜色、没有预着色顶点且只有双色棍棒的情况下仍然成立。然后，我们针对完全未着色的实例给出一个简单的确定性$\ rac{2}{c(c+1)}$近似算法，当所有棍棒都是双色时，该比率改进为$\ rac{2}{c(c-1)}$。接下来，我们通过将剩余的着色选择归约为在划分拟阵上的单调子模最大化，获得针对具有$c$种颜色的任意实例的随机化$\ rac{1-\ rac{1}{e}}{2c}$近似算法。在二分图上，近似比率改进为$\ rac{1-\ rac{1}{e}}{c}$。对于每个固定的$c$，我们进一步在一般图和二分图上分别获得确定性的$\ rac{1}{2c}$和$\ rac{1}{c}$近似算法，时间复杂度为$n^{O(c^2)}$。最后，对于每个固定的颜色数量，我们证明\ extsc{MaxMMP}可以在树和余图上以$n^{O(c^2)}$的时间精确求解。

英文摘要

In this paper, we study \textsc{MaxMMP}, an optimization variant of the Matching-Match Puzzle introduced by Iburi and Uehara (FUN 2024). Given a graph, a partial vertex coloring, and a multiset of colored sticks, the goal is to complete the coloring and assign the sticks to graph edges so as to maximize the number of satisfied edges. We first prove that \textsc{MaxMMP} is APX-hard by an reduction from \textsc{Max-Cut}. The hardness already holds with two colors, no precolored vertices, and only bichromatic sticks. We then give a simple deterministic $\frac{2}{c(c+1)}$-approximation for completely uncolored instances, improving to $\frac{2}{c(c-1)}$ when all sticks are bichromatic. Next, we obtain a randomized $\frac{1-\frac{1}{e}}{2c}$-approximation for arbitrary instances with $c$ colors by reducing the remaining coloring choices to monotone submodular maximization under a partition matroid. On bipartite graphs, the approximation ratio improves to $\frac{1-\frac{1}{e}}{c}$. For every fixed $c$, we further obtain deterministic $\frac{1}{2c}$ and $\frac{1}{c}$-approximations on general and bipartite graphs, respectively, in time $n^{O(c^2)}$. Finally, for every fixed number of colors, we show that \textsc{MaxMMP} can be solved exactly in time $n^{O(c^2)}$ on trees and on cographs.

发表机构

• University of Bucharest（布加勒斯特大学）
• National Institute for Research and Development in Informatics（信息技术研究与开发国家研究所）

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