The Kneser graph $K(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $[n]=\{1,2,\ldots,n\}$ where two such sets are adjacent if they are disjoint. The Schrijver graph $S(n,k)$ is defined as the subgraph of $K(n,k)$ induced by the collection of all $k$-subsets of $[n]$ that do not include two consecutive elements modulo $n$. It is known that the chromatic number of both $K(n,k)$ and $S(n,k)$ is $n-2k+2$. In the computational Kneser and Schrijver problems, we are given an access to a coloring with $n-2k+1$ colors of the vertices of $K(n,k)$ and $S(n,k)$ respectively, and the goal is to find a monochromatic edge. We prove that the problems admit randomized algorithms with running time $n^{O(1)} \cdot k^{O(k)}$, hence they are fixed-parameter tractable with respect to the parameter $k$. The analysis involves structural results on intersecting families and on induced subgraphs of Kneser and Schrijver graphs. We also study the Agreeable-Set problem of assigning a small subset of a set of $m$ items to a group of $\ell$ agents, so that all agents value the subset at least as much as its complement. As an application of our algorithm for the Kneser problem, we obtain a randomized polynomial-time algorithm for the Agreeable-Set problem for instances with $\ell \geq m - O(\frac{\log m}{\log \log m})$. We further show that the Agreeable-Set problem is at least as hard as a variant of the Kneser problem with an extended access to the input coloring.

翻译：Kneser图 $K(n,k)$ 定义于满足 $n \geq 2k$ 的整数 $n$ 和 $k$，其顶点为 $[n]=\{1,2,\ldots,n\}$ 的所有 $k$ 元子集，当两个子集互不相交时，它们在图中有边相连。Schrijver图 $S(n,k)$ 定义为 $K(n,k)$ 的子图，由 $[n]$ 中不含模 $n$ 意义下连续两个元素的所有 $k$ 元子集诱导生成。已知 $K(n,k)$ 和 $S(n,k)$ 的色数均为 $n-2k+2$。在计算Kneser与Schrijver问题中，我们分别获得对 $K(n,k)$ 和 $S(n,k)$ 顶点使用 $n-2k+1$ 种颜色的着色访问，目标是找到一条单色边。我们证明这些问题存在运行时间为 $n^{O(1)} \cdot k^{O(k)}$ 的随机化算法，因此关于参数 $k$ 是固定参数易处理的。分析过程涉及交族的结构性质以及Kneser图与Schrijver图的诱导子图。我们还研究了Agreeable-Set问题：将 $m$ 个物品中的一个小子集分配给 $\ell$ 个代理，使得所有代理对该子集的估值均不低于其补集。作为Kneser问题算法的一个应用，我们对于满足 $\ell \geq m - O(\frac{\log m}{\log \log m})$ 的实例获得了Agreeable-Set问题的随机化多项式时间算法。我们进一步证明，Agreeable-Set问题至少与一种具有扩展输入着色访问的Kneser问题变体同样困难。