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.

中文翻译：

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在循环中寻找公平独立集的复杂性

令 $G$ 为循环图，令 $V_1,\ldots,V_m$ 为其顶点集划分为 $m$ 集。如果$|S \cap V_i|，则称$G$ 的独立集合$S$ 公平地代表了分区。\geq \frac{1}{2} \cdot |V_i| -1$ 为所有 $i \in [m]$。众所周知，对于每个循环及其顶点集的每个分区，都存在一个独立的集合来公平地表示分区（Aharoni 等人，离散数学之旅，2017）。我们证明找到这样一个独立集的问题是 $\mathsf{PPA}$-complete。作为一个应用，我们展示了在 Schrijver 图中找到单色边的问题，给定使用比其色数更少的颜色的着色的简洁表示，也是 $\mathsf{PPA}$-complete。这项工作的动机是“循环加三角形”的计算方面