High-Performance Parallel Graph Coloring with Strong Guarantees on Work, Depth, and Quality

TL;DR

This paper introduces parallel graph coloring heuristics with strong theoretical guarantees on work, depth, and coloring quality, achieving efficient and high-quality colorings for large graphs.

Contribution

It presents a novel relaxation of the degeneracy order enabling parallelization with provable guarantees and superior coloring quality.

Findings

Ensures polylogarithmic depth in coloring algorithms.

Achieves better color bounds than existing parallel schemes.

Offers competitive runtime and superior coloring quality on real-world graphs.

Abstract

We develop the first parallel graph coloring heuristics with strong theoretical guarantees on work and depth and coloring quality. The key idea is to design a relaxation of the vertex degeneracy order, a well-known graph theory concept, and to color vertices in the order dictated by this relaxation. This introduces a tunable amount of parallelism into the degeneracy ordering that is otherwise hard to parallelize. This simple idea enables significant benefits in several key aspects of graph coloring. For example, one of our algorithms ensures polylogarithmic depth and a bound on the number of used colors that is superior to all other parallelizable schemes, while maintaining work-efficiency. In addition to provable guarantees, the developed algorithms have competitive run-times for several real-world graphs, while almost always providing superior coloring quality. Our degeneracy ordering relaxation is of separate interest for algorithms outside the context of coloring.