The Complexity of Finding Fair Independent Sets in Cycles
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Let G be a cycle graph and let V_1,…,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 ∩ V_i| ≥1/2· |V_i| -1 for all i ∈ [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 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.
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