Fast Deterministic Chromatic Number under the Asymptotic Rank Conjecture

TL;DR

This work shows that, assuming Strassen's asymptotic rank conjecture, one can deterministically compute the chromatic number of an $n$-vertex graph in $O(1.99982^n)$ time by combining balanced coloring techniques with a derandomized three-way partitioning framework. The authors extend Pratt's tensor-based approach to the unbalanced case and generalize it to large-subset set cover, deriving near-optimal deterministic runtimes for coloring and related problems. The key technical contributions include ν-balanced $k$-covers detection, a block-balanced partitioning method, and a balancing family construction via pairwise independent hashing, all under the AR conjecture. The results illuminate a conditional barrier between $2^n$-time and subexponential deterministic algorithms for chromatic number, with potential implications for fine-grained complexity and tensor-based algorithm design.

Abstract

In this paper we further explore the recently discovered connection by Björklund 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 Björklund, 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.

Figures (2)

• Figure 1: The partitioning tensors $T_{1/3,\,6}$ and $T_{1/3,\,9}$.
• Figure 2: On the left, a three-way partitioning $A_1 \,\dot{\cup}\, A_2 \,\dot{\cup}\, A_3 = [n]$ is shown horizontally. On the right, the universe $[n]$ is vertically partitioned into blocks $B_1, \ldots , B_s$. Each set $A_i$ for $i\in[3]$ is $(\delta,r)$-balanced onto each block.

Theorems & Definitions (31)

• Theorem 1: Main; Deterministic fine-grained three-way partitioning
• Theorem 2: Set cover with large subsets under the asymptotic rank conjecture
• Theorem 3: Chromatic number under the asymptotic rank conjecture
• Conjecture 4: Asymptotic rank conjecture Strassen1994BurgisserCS2013
• Lemma 5
• proof
• Lemma 6
• proof
• Claim 7
• proof