Fast Deterministic Chromatic Number under the Asymptotic Rank Conjecture

TL;DR

This paper demonstrates that under the asymptotic rank conjecture, the chromatic number can be computed deterministically in sub-exponential time, linking graph coloring complexity to tensor rank conjectures and providing conditional bounds.

Contribution

It establishes a deterministic algorithm for chromatic number under the asymptotic rank conjecture and explores implications for tensor rank and algorithmic complexity.

Findings

Deterministic $O(1.99982^n)$ time algorithm for chromatic number under the conjecture

Conditional falsification of the asymptotic rank conjecture if chromatic number is hard to compute

Extension of Pratt's tensor algorithm to unbalanced three-way partitioning

Abstract

In this paper we further explore the recently discovered connection by Bj\"{o}rklund and Kaski [STOC 2024] and Pratt [STOC 2024] between the asymptotic rank conjecture of Strassen [Progr. Math. 1994] and the three-way partitioning problem. We show that under the asymptotic rank conjecture, the chromatic number of an $n$-vertex graph can be computed deterministically in $O(1.99982^n)$ time, thus giving a conditional answer to a question of Zamir [ICALP 2021], and questioning the optimality of the $2^n\operatorname{poly}(n)$ time algorithm for chromatic number by Bj\"{o}rklund, Husfeldt, and Koivisto [SICOMP 2009]. Viewed in the other direction, if chromatic number indeed requires deterministic algorithms to run in close to $2^n$ time, we obtain a sequence of explicit tensors of superlinear rank, falsifying the asymptotic rank conjecture. Our technique is a combination of earlier algorithms for detecting $k$-colorings for small $k$ and enumerating $k$-colorable subgraphs, with an extension and derandomisation of Pratt's tensor-based algorithm for balanced three-way partitioning to the unbalanced case.