A Grover search-based algorithm for the list coloring problem

TL;DR

This paper introduces a quantum algorithm based on Grover search to improve the efficiency of solving the list coloring problem in graphs, especially for arbitrary instances, offering a quadratic speedup over classical exhaustive search.

Contribution

The paper presents a novel quantum algorithm leveraging Grover search to address the list coloring problem, extending quantum approaches to more general graph coloring scenarios.

Findings

Quadratic speedup over classical exhaustive search for list coloring.

Algorithm performs better in arbitrary graph and list cases.

Classical algorithms outperform in restricted problem instances.

Abstract

Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for $k$-coloring of graphs on $n$ vertices has runtimes $\Omega(2^n)$ for $k\ge 5$. The list coloring problem asks the following more general question: given a list of available colors for each vertex in a graph, does it admit a proper coloring? We propose a quantum algorithm based on Grover search to quadratically speed up exhaustive search. Our algorithm loses in complexity to classical ones in specific restricted cases, but improves exhaustive search for cases where the lists and graphs considered are arbitrary in nature.