A Fixed-Parameter Algorithm for the Kneser Problem

TL;DR

This paper introduces a fixed-parameter randomized algorithm for the Kneser problem, demonstrating its tractability for fixed $k$, and applies it to develop efficient algorithms for the Agreeable-Set problem under certain conditions.

Contribution

It presents the first fixed-parameter algorithm for the Kneser problem and connects it to the Agreeable-Set problem, expanding the algorithmic understanding of these combinatorial problems.

Findings

The Kneser problem is fixed-parameter tractable with respect to $k$.

The algorithm runs in $n^{O(1)} imes k^{O(k)}$ time, showing practical efficiency for small $k$.

A polynomial-time randomized algorithm for the Agreeable-Set problem is achieved under specific parameters.

Abstract

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 $\{1,2,\ldots,n\}$ where two such sets are adjacent if they are disjoint. A classical result of Lov\'asz asserts that the chromatic number of $K(n,k)$ is $n-2k+2$. In the computational Kneser problem, we are given an oracle access to a coloring of the vertices of $K(n,k)$ with $n-2k+1$ colors, and the goal is to find a monochromatic edge. We present a randomized algorithm for the Kneser problem with running time $n^{O(1)} \cdot k^{O(k)}$. This shows that the problem is fixed-parameter tractable with respect to the parameter $k$. The analysis involves structural results on intersecting families and on induced subgraphs of Kneser 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 that satisfy $\ell \geq m - O(\frac{\log m}{\log \log m})$. 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.