Edge-Coloring Algorithms for Bounded Degree Multigraphs

TL;DR

This paper introduces improved algorithms for edge-coloring multigraphs with bounded degree, achieving faster sequential and distributed solutions, and extending existing methods to multigraphs based on Shannon's theorem.

Contribution

It presents the first $O(n ext{log} n)$ sequential algorithm and near-optimal randomized algorithm, along with distributed algorithms, for edge-coloring multigraphs, extending previous simple graph results.

Findings

Sequential algorithms run in $O(n\log n)$ and $O(n)$ time.

Distributed algorithms run in $ ilde O(\log^5 n)$ and $O(\log^2 n)$ rounds.

Algorithms are based on entropy compression and extend Vizing's theorem to multigraphs.

Abstract

In this paper, we consider algorithms for edge-coloring multigraphs $G$ of bounded maximum degree, i.e., $\Delta(G) = O(1)$. Shannon's theorem states that any multigraph of maximum degree $\Delta$ can be properly edge-colored with $\lfloor3\Delta/2\rfloor$ colors. Our main results include algorithms for computing such colorings. We design deterministic and randomized sequential algorithms with running time $O(n\log n)$ and $O(n)$, respectively. This is the first improvement since the $O(n^2)$ algorithm in Shannon's original paper, and our randomized algorithm is optimal up to constant factors. We also develop distributed algorithms in the $\mathsf{LOCAL}$ model of computation. Namely, we design deterministic and randomized $\mathsf{LOCAL}$ algorithms with running time $\tilde O(\log^5 n)$ and $O(\log^2n)$, respectively. The deterministic sequential algorithm is a simplified extension of earlier work of Gabow et al. in edge-coloring simple graphs. The other algorithms apply the entropy compression method in a similar way to recent work by the author and Bernshteyn, where the authors design algorithms for Vizing's theorem for simple graphs. We also extend those results to Vizing's theorem for multigraphs.