On Finding Constrained Independent Sets in Cycles

A subset of \([n] = \{1,2,\ldots ,n\}\) is called stable if it forms an independent set in the cycle on the vertex set [n]. In 1978, Schrijver proved via a topological argument that for all integers n and k with \(n \ge 2k\), the family of stable k-subsets of [n] cannot be covered by \(n-2k+1\) intersecting families. We study two total search problems whose totality relies on this result. In the first problem, denoted by \(\textsc {Schrijver}(n,k,m)\), we are given an access to a coloring of the stable k-subsets of [n] with \(m = m(n,k)\) colors, where \(m \le n-2k+1\), and the goal is to find a pair of disjoint subsets that are assigned the same color. While for \(m = n-2k+1\) the problem is known to be \(\textsf{PPA}\)-complete, we prove that for \(m < d \cdot \lfloor \frac{n}{2k+d-2} \rfloor \), with d being any fixed constant, the problem admits an efficient algorithm. For \(m = \lfloor n/2 \rfloor -2k+1\), we prove that the problem is efficiently reducible to the \(\textsc {Kneser}\) problem. Motivated by the relation between the problems, we investigate the family of unstable k-subsets of [n], which might be of independent interest. In the second problem, called Unfair Independent Set in Cycle, we are given \(\ell \) subsets \(V_1, \ldots , V_\ell \) of [n], where \(\ell \le n-2k+1\) and \(|V_i| \ge 2\) for all \(i \in [\ell ]\), and the goal is to find a stable k-subset S of [n] satisfying the constraints \(|S \cap V_i| \le |V_i|/2\) for \(i \in [\ell ]\). We prove that the problem is \(\textsf{PPA}\)-complete and that its restriction to instances with \(n=3k\) is at least as hard as the Cycle plus Triangles problem, for which no efficient algorithm is known. On the contrary, we prove that there exists a constant c for which the restriction of the problem to instances with \(n \ge c \cdot k\) can be solved in polynomial time.