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 $n \geq 2k$, 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 $\mathsf{Schrijve}r(n,k,m)$, we are given an access to a coloring of the stable $k$-subsets of $[n]$ with $m = m(n,k)$ colors, where $m \leq n-2k+1$, 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 $\mathsf{PPA}$-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 $\mathsf{Kneser}$ 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 $\ell \leq n-2k+1$ and $|V_i| \geq 2$ for all $i \in [\ell]$, and the goal is to find a stable $k$-subset $S$ of $[n]$ satisfying the constraints $|S \cap V_i| \leq |V_i|/2$ for $i \in [\ell]$. We prove that the problem is $\mathsf{PPA}$-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 $n \geq c \cdot k$ can be solved in polynomial time.

翻译：$[n] = \{1,2,\ldots,n\}$的子集如果构成顶点集$[n]$上环的独立集，则称为稳定子集。1978年，Schrijver通过拓扑论证证明：对于所有满足$n \geq 2k$的整数$n$和$k$，$[n]$的稳定$k$-子集族无法被$n-2k+1$个相交族覆盖。我们研究两个依赖该结论的总搜索问题。在第一个问题（记为$\mathsf{Schrijver}(n,k,m)$）中，给定对$[n]$的稳定$k$-子集的一种着色，颜色数为$m = m(n,k)$且$m \leq n-2k+1$，目标是找到被分配相同颜色的一对不相交子集。当$m = n-2k+1$时该问题已知为$\mathsf{PPA}$-完全的，但我们证明：对于$m < d \cdot \lfloor \frac{n}{2k+d-2} \rfloor$（其中$d$为任意固定常数），该问题存在高效算法。当$m = \lfloor n/2 \rfloor-2k+1$时，我们证明该问题可高效归约至$\mathsf{Kneser}$问题。受两者关联启发，我们研究了$[n]$的不稳定$k$-子集族，该族可能具有独立研究价值。在第二个问题（称为环中不公平独立集）中，给定$[n]$的$\ell$个子集$V_1, \ldots, V_\ell$，其中$\ell \leq n-2k+1$且对所有$i \in [\ell]$有$|V_i| \geq 2$，目标是找到$[n]$的一个稳定$k$-子集$S$，使得对任意$i \in [\ell]$满足约束$|S \cap V_i| \leq |V_i|/2$。我们证明该问题是$\mathsf{PPA}$-完全的，且限制到$n=3k$的实例至少与环加三角形问题（目前尚无已知高效算法）同样困难。相反地，我们证明存在常数$c$，使得限制到$n \geq c \cdot k$的实例可在多项式时间内求解。