求圈中公平独立集的复杂性
The Complexity of Finding Fair Independent Sets in Cycles
摘要：
设$G$是一个圈图，$V_1,\ldots,V_m$是它的顶点集到$m$集合的一个划分.如果所有$i \in [m]$的$|S \cap V_i| \geq \frac{1}{2} \cdot |V_i| -1$都是$G$的独立集$S$，则可以公平地表示该分区。已知对于其顶点集的每个循环和每个划分，存在公平地表示该划分的独立集（Aharoni等人，离散数学之旅2017年）。我们证明了寻找这样一个独立集的问题是$\mathsf{PPA}$ -完全的。作为一个应用程序，我们发现，在Schrijver图中找到一个单色的边缘，给出了一个简洁的表示着色，使用较少的颜色比它的色数的问题，是$\mathsf{PPA}$，以及完成。这项工作的动机是计算方面的“循环加三角形”的问题及其扩展。
英文摘要：
Let $G$ be a cycle graph and let $V_1,\ldots,V_m$ be a partition of its vertex set into $m$ sets. An independent set $S$ of $G$ is said to fairly represent the partition if $|S \cap V_i| \geq \frac{1}{2} \cdot |V_i| -1$ for all $i \in [m]$. It is known that for every cycle and every partition of its vertex set, there exists an independent set that fairly represents the partition (Aharoni et al., A Journey through Discrete Math., 2017). We prove that the problem of finding such an independent set is $\mathsf{PPA}$-complete. As an application, we show that the problem of finding a monochromatic edge in a Schrijver graph, given a succinct representation of a coloring that uses fewer colors than its chromatic number, is $\mathsf{PPA}$-complete as well. The work is motivated by the computational aspects of the `cycle plus triangles' problem and of its extensions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
