The Complexity of Finding Fair Independent Sets in Cycles

Abstract

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. 2017). We prove that the problem of finding such an independent set is 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 PPA-complete as well. The work is motivated by the computational aspects of the `cycle plus triangles' problem and of its extensions.