Improved Sublinear Algorithms for Classical and Quantum Graph Coloring

TL;DR

This paper introduces three new sublinear algorithms for graph coloring, including classical and quantum approaches, that improve runtime and coloring bounds for graphs with maximum degree .

Contribution

It presents novel classical and quantum sublinear algorithms for vertex coloring, improving runtime and coloring bounds over previous methods.

Findings

Classical algorithm achieves +1 colors with faster runtime.

Quantum algorithms further accelerate coloring with Grover's search.

New quantum algorithm for near-optimal -coloring improves previous bounds.

Abstract

We present three sublinear randomized algorithms for vertex-coloring of graphs with maximum degree $\Delta$. The first is a simple algorithm that extends the idea of Morris and Song to color graphs with maximum degree $\Delta$ using $\Delta+1$ colors. Combined with the greedy algorithm, it achieves an expected runtime of $O(n^{3/2}\sqrt{\log n})$ in the query model, improving on Assadi, Chen, and Khanna's algorithm by a $\sqrt{\log n}$ factor in expectation. When we allow quantum queries to the graph, we can accelerate the first algorithm using Grover's famous algorithm, resulting in a runtime of $\tilde{O}(n^{4/3})$ quantum queries. Finally, we introduce a quantum algorithm for $(1+\epsilon)\Delta$-coloring, achieving $O(\epsilon^{-1}n^{5/4}\log^{3/2}n)$ quantum queries, offering a polynomial improvement over the previous best bound by Morris and Song.