Towards Optimal Distributed Edge Coloring with Fewer Colors

Abstract

There is a huge difference in techniques and runtimes of distributed algorithms for problems that can be solved by a sequential greedy algorithm and those that cannot. A prime example of this contrast appears in the edge coloring problem: while $(2\Delta-1)$-edge coloring can be solved in $\mathcal{O}(\log^{\ast}(n))$ rounds on constant-degree graphs, the seemingly minor reduction to $(2\Delta-2)$ colors leads to an $\Omega(\log n)$ lower bound [Chang, He, Li, Pettie & Uitto, SODA'18]. Understanding this sharp divide between very local problems and inherently more global ones remains a central open question in distributed computing and it is a core focus of this paper. As our main contribution we design a deterministic distributed $\mathcal{O}(\log n)$-round reduction from the $(2\Delta-2)$-edge coloring problem to the much easier $(2\Delta-1)$-edge coloring problem. This reduction is optimal, as the $(2\Delta-2)$-edge coloring problem admits an $\Omega(\log n)$ lower bound, whereas the $2\Delta-1$-edge coloring problem can be solved in $\mathcal{O}(\log^{\ast}n)$ rounds. By plugging in the $(2\Delta-1)$-edge coloring algorithms from [Balliu, Brandt, Kuhn & Olivetti, PODC'22] running in $\mathcal{O}(\log^{12}\Delta + \log^{\ast} n)$ rounds, we obtain an optimal runtime of $\mathcal{O}(\log n)$ rounds as long as $\Delta = 2^{\mathcal{O}(\log^{1/12} n)}$. Furthermore, on general graphs our reduction improves the runtime from $\widetilde{\mathcal{O}}(\log^3 n)$ to $\widetilde{\mathcal{O}}(\log^{5/3} n)$. In addition, we also obtain an optimal $\mathcal{O}(\log \log n)$-round randomized reduction of $(2\Delta - 2)$-edge coloring to $(2\Delta - 1)$-edge coloring. Lastly, we obtain an $\mathcal{O}(\log_\Delta n)$-round reduction from the $(2\Delta-1)$-edge coloring, albeit to the somewhat harder maximal independent set (MIS) problem.