$ \renewcommand{\tilde}{\widetilde} $We present an $\tilde{O}(\log^2 n)$ round deterministic distributed algorithm for the maximal independent set problem. By known reductions, this round complexity extends also to maximal matching, $\Delta+1$ vertex coloring, and $2\Delta-1$ edge coloring. These four problems are among the most central problems in distributed graph algorithms and have been studied extensively for the past four decades. This improved round complexity comes closer to the $\tilde{\Omega}(\log n)$ lower bound of maximal independent set and maximal matching [Balliu et al. FOCS '19]. The previous best known deterministic complexity for all of these problems was $\Theta(\log^3 n)$. Via the shattering technique, the improvement permeates also to the corresponding randomized complexities, e.g., the new randomized complexity of $\Delta+1$ vertex coloring is now $\tilde{O}(\log^2\log n)$ rounds. Our approach is a novel combination of the previously known two methods for developing deterministic algorithms for these problems, namely global derandomization via network decomposition (see e.g., [Rozhon, Ghaffari STOC'20; Ghaffari, Grunau, Rozhon SODA'21; Ghaffari et al. SODA'23]) and local rounding of fractional solutions (see e.g., [Fischer DISC'17; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Ghaffari, Kuhn FOCS'21; Faour et al. SODA'23]). We consider a relaxation of the classic network decomposition concept, where instead of requiring the clusters in the same block to be non-adjacent, we allow each node to have a small number of neighboring clusters. We also show a deterministic algorithm that computes this relaxed decomposition faster than standard decompositions. We then use this relaxed decomposition to significantly improve the integrality of certain fractional solutions, before handing them to the local rounding procedure that now has to do fewer rounding steps.

翻译：摘要: 我们提出一个运行轮数为$\tilde{O}(\log^2 n)$的确定性分布式算法，用于求解最大独立集问题。通过已知的归约，该轮复杂度同样适用于最大匹配、$\Delta+1$顶点着色以及$2\Delta-1$边着色问题。这四个问题属于分布式图算法中最核心的问题，并在过去四十年间得到了广泛研究。这一轮复杂度的提升更接近最大独立集和最大匹配的$\tilde{\Omega}(\log n)$下界[Balliu等人，FOCS '19]。此前这些问题的已知最优确定性复杂度均为$\Theta(\log^3 n)$。通过碎裂技术，该改进也渗透至相应的随机化复杂度，例如$\Delta+1$顶点着色问题的最新随机化复杂度降至$\tilde{O}(\log^2\log n)$轮。我们的方法创新性地结合了此前两种用于开发这些问题的确定性算法的技术，即通过网络分解的全局去随机化（参见例如[Rozhon, Ghaffari STOC'20; Ghaffari, Grunau, Rozhon SODA'21; Ghaffari等人 SODA'23]）和分数解的局部舍入（参见例如[Fischer DISC'17; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Ghaffari, Kuhn FOCS'21; Faour等人 SODA'23]）。我们考虑了经典网络分解概念的一个松弛版本，其中不要求同一区块中的簇彼此不相邻，而是允许每个节点拥有少量相邻簇。我们还展示了一个确定性算法，能够比标准分解更快地计算出这种松弛分解。随后，我们利用这种松弛分解显著改善了某些分数解的完整性，再将它们交给现在只需执行较少舍入步骤的局部舍入过程。