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.

翻译：在分布式计算的LOCAL模型中，基础图问题的确定性复杂度近期取得了一系列进展：最大独立集（MIS）和$(\Delta + 1)$-着色问题已获得$\tilde{O}(\log^{5/3} n)$轮的最先进上界[Ghaffari, Grunau, FOCS'24]，而限制更强的$\Delta$-着色问题则达到$\tilde{O}(\log^{19/9} n)$轮上界[Ghaffari, Kuhn, FOCS'21; Ghaffari, Grunau, FOCS'24; Bourreau, Brandt, Nolin, STOC'25]。本文证明$\Delta$-着色问题同样可以在$\tilde{O}(\log^{5/3} n)$轮内确定性求解，从而匹配当前$(\Delta + 1)$-着色的最佳上界。我们通过构建从$\Delta$-着色到MIS的归约实现该结果，该归约保证：除非MIS能在亚对数时间内求解（此时根据[BFHKLRSU, STOC'16]的$\Omega(\log n)$轮$\Delta$-着色下界，我们的归约将导出$\Delta$-着色的紧确复杂度$\Theta(\log n)$），否则$\Delta$-着色的（渐近）复杂度至多为MIS的复杂度。特别地，MIS复杂度的任何改进都将直接转化为$\Delta$-着色复杂度的同等改进（直至$\Delta$-着色的真实复杂度）。我们的归约在随机化LOCAL模型以及同时以$n$和$\Delta$为参数的复杂度分析中均带来改进：在一般图上获得$\tilde{O}(\log^{5/3} \log n)$轮的随机化复杂度上界（改进当前最优结果$\tilde{O}(\log^{8/3} \log n)$轮），在有界度图上分别获得确定性$\Theta(\log n)$与随机化$\Theta(\log \log n)$的紧确复杂度。对于常数团数图类（例如二分图），我们还给出了到$(\Delta+1)$-着色问题的归约。