Improved Sublinear Algorithms for Classical and Quantum Graph Coloring

TL;DR

The paper studies sublinear quantum and classical algorithms for coloring graphs with maximum degree $\Delta$, focusing on reducing query complexity in the adjacency- and neighborhood-query models. It introduces a classical Las Vegas algorithm coloring with $(\Delta+1)$ colors using $O(n^2\log n/\Delta)$ adjacency queries, and combines it with a greedy step to reach $O(n^{3/2}\sqrt{\log n})$ queries in expectation. In the quantum setting, it leverages Grover search to obtain sublinear quantum colorings, proving an $O(n^{3/2}\log n/\sqrt{\Delta})$ adjacency-query bound and a $\tilde{O}(n^{4/3})$ quantum-query bound for the same goal, effectively beating classical lower bounds. It also provides near-optimal quantum approaches for $(1+\varepsilon)\Delta$-coloring with two variants: a $O(\varepsilon^{-2} n \log^2 n \sqrt{\Delta})$ neighborhood-query algorithm (success prob. $\ge 2/3$) and a $\tilde{O}(\varepsilon^{-1} n^{5/4})$-query algorithm, both offering polynomial improvements over prior work. Together, these results advance sublinear colorings in both classical and quantum regimes and open avenues for improved partitioning and cross-model methods.

Abstract

We present three sublinear randomized algorithms for vertex-coloring of graphs with maximum degree $Δ$. The first is a simple algorithm that extends the idea of Morris and Song to color graphs with maximum degree $Δ$ using $Δ+1$ 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 $(1+ε)Δ$-coloring, achieving $O(ε^{-1}n^{5/4}\log^{3/2}n)$ quantum queries, offering a polynomial improvement over the previous best bound by Morris and Song.

Theorems & Definitions (14)

• Theorem 1
• Corollary 2
• Theorem 3
• Corollary 4
• Theorem 5
• Corollary 6
• proof : Proof of Theorem \ref{['thm: classical Delta+1']}
• Theorem 7: Grover's Algorithm boyer1998tight
• proof : Proof of Theorem \ref{['thm: quantum adjacency']}
• Lemma 8