Faster Distributed $\Delta$-Coloring via a Reduction to MIS

Abstract

Recent improvements on the deterministic complexities of fundamental graph problems in the LOCAL model of distributed computing have yielded state-of-the-art upper bounds of $\tilde{O}(\log^{5/3} n)$ rounds for maximal independent set (MIS) and $(\Delta + 1)$-coloring [Ghaffari, Grunau, FOCS'24] and $\tilde{O}(\log^{19/9} n)$ rounds for the more restrictive $\Delta$-coloring problem [Ghaffari, Kuhn, FOCS'21; Ghaffari, Grunau, FOCS'24; Bourreau, Brandt, Nolin, STOC'25]. In our work, we show that $\Delta$-coloring can be solved deterministically in $\tilde{O}(\log^{5/3} n)$ rounds as well, matching the currently best bound for $(\Delta + 1)$-coloring. We achieve our result by developing a reduction from $\Delta$-coloring to MIS that guarantees that the (asymptotic) complexity of $\Delta$-coloring is at most the complexity of MIS, unless MIS can be solved in sublogarithmic time, in which case, due to the $\Omega(\log n)$-round $\Delta$-coloring lower bound from [BFHKLRSU, STOC'16], our reduction implies a tight complexity of $\Theta(\log n)$ for $\Delta$-coloring. In particular, any improvement on the complexity of the MIS problem will yield the same improvement for the complexity of $\Delta$-coloring (up to the true complexity of $\Delta$-coloring). Our reduction yields improvements for $\Delta$-coloring in the randomized LOCAL model and when complexities are parameterized by both $n$ and $\Delta$. We obtain a randomized complexity bound of $\tilde{O}(\log^{5/3} \log n)$ rounds (improving over the state of the art of $\tilde{O}(\log^{8/3} \log n)$ rounds) on general graphs and tight complexities of $\Theta(\log n)$ and $\Theta(\log \log n)$ for the deterministic, resp.\ randomized, complexity on bounded-degree graphs. In the special case of graphs of constant clique number (which for instance include bipartite graphs), we also give a reduction to the $(\Delta+1)$-coloring problem.