Breaking the $2^n$ barrier for graph $k$-coloring

2026-07-29

AI Summary

The $k$-coloring problem on general graphs has long lacked a universal algorithm surpassing the $2^n$ time barrier. This work proposes a randomized one-sided error algorithm that, for the first time, uniformly breaks this barrier for all $k \geq 3$, extending exponential speedups previously known only for $k \leq 6$ to arbitrary values of $k$. Through a novel randomized design and refined exponential-time analysis, the authors establish the existence of a constant $\varepsilon_k > 0$ such that the problem can be solved in $O((2 - \varepsilon_k)^n)$ time, significantly improving upon the classical bound.

Abstract

We show that for all $k$, there exists $\varepsilon_k > 0$ such that graph $k$-coloring can be solved by a randomized algorithm with one-sided error in time $O((2-\varepsilon_k)^n)$. Prior to this work and independent concurrent work of Zamir [arXiv, 2026], exponential improvements over the $2^n \cdot \mathrm{poly}(n)$-time algorithm of Björklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for $k \le 6$.

Problem

• graph k-coloring
• exponential time algorithms
• time complexity
• randomized algorithms
• fine-grained complexity

Innovation

• graph k-coloring
• exponential-time algorithm
• randomized algorithm
• time complexity
• fine-grained complexity