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.

翻译：本文进一步探讨了Björklund与Kaski [STOC 2024] 以及Pratt [STOC 2024] 近期发现的Strassen渐近秩猜想 [Progr. Math. 1994] 与三路划分问题之间的关联。我们证明在渐近秩猜想成立的条件下，$n$顶点图的色数可以在$O(1.99982^n)$时间内被确定性计算，从而为Zamir [ICALP 2021] 的问题提供了条件性解答，并对Björklund、Husfeldt与Koivisto [SICOMP 2009] 提出的$2^n\operatorname{poly}(n)$时间色数算法的最优性提出了质疑。从反向视角来看，若色数计算确实要求确定性算法的运行时间接近$2^n$，我们将得到一系列具有超线性秩的显式张量，从而证伪渐近秩猜想。我们的技术结合了早期检测小$k$值着色与枚举$k$可着色子图的算法，并将Pratt基于张量的平衡三路划分算法扩展并去随机化至非平衡情形。