The Complexity of Finding Fair Independent Sets in Cycles

TL;DR

This paper proves that finding a fair independent set in cycle graphs, which always exists, is computationally hard (PPA-complete), and applies this result to coloring problems in Schrijver graphs.

Contribution

It establishes the PPA-completeness of the problem of finding fair independent sets in cycles, revealing its computational complexity.

Findings

Finding fair independent sets in cycles is PPA-complete.

The problem of finding monochromatic edges in Schrijver graphs is also PPA-complete.

The work connects cycle problems with graph coloring complexity.

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., 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.