Deterministic Algorithm for Computing Chromatic Number in 1.9999^n Time under the Asymptotic Rank Conjecture

Core Concepts

Under the asymptotic rank conjecture, the chromatic number of an n-vertex graph can be computed deterministically in O(1.99982^n) time.

Abstract

The paper presents a deterministic algorithm for computing the chromatic number of an n-vertex graph in O(1.99982^n) time, assuming the asymptotic rank conjecture is true. The key insights are: The authors show that under the asymptotic rank conjecture, the three-way partitioning problem can be solved deterministically in near-optimal time. This allows them to efficiently detect balanced k-colorings of a graph. To handle unbalanced colorings, the authors combine their three-way partitioning algorithm with existing deterministic algorithms for 4-coloring. This allows them to cover all possible colorings and compute the chromatic number. The authors leverage the connection between set cover problems and graph coloring to extend their results from balanced colorings to the general chromatic number problem. The paper also includes results on deterministic set cover algorithms under the asymptotic rank conjecture, which serve as building blocks for the chromatic number algorithm.

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Key Insights Distilled From

Chromatic number in $1.9999^n$ time? Fast deterministic set partitioning under the asymptotic rank conjecture

by Andr... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04987.pdf

Deeper Inquiries

What other graph problems could potentially be solved efficiently under the asymptotic rank conjecture

The asymptotic rank conjecture has the potential to impact the efficiency of solving various graph problems beyond the chromatic number. One such problem is the maximum independent set problem. By leveraging the insights and algorithms developed for the chromatic number problem, it is possible to explore deterministic algorithms for finding maximum independent sets in graphs. The connection between tensor operations and graph properties opens up avenues for developing faster algorithms for problems like maximum independent sets, vertex cover, and even problems related to graph connectivity.

How tight is the O(1.99982^n) running time bound for the chromatic number algorithm

The O(1.99982^n) running time bound for the chromatic number algorithm is quite tight and represents a significant improvement over previous algorithms. While the constant in the exponent is already optimized to a large extent, there may still be room for minor improvements through algorithmic optimizations and fine-tuning of the techniques used. However, achieving a substantial reduction in the exponent constant may be challenging due to the inherent complexity of the problem and the underlying mathematical structures involved.

Is it possible to further improve the constant in the exponent

The implications of the asymptotic rank conjecture extend beyond graph algorithms to various fundamental problems in computer science. The conjecture suggests a fundamental limit on the complexity of tensor operations, which are prevalent in many computational tasks. If the conjecture holds true, it could lead to breakthroughs in areas such as algebraic complexity theory, computational linear algebra, and even machine learning algorithms that rely on tensor computations. The conjecture has the potential to revolutionize the way we approach and analyze complex computational problems across different domains.