分布式$Δ$ -着色玩捉迷藏
Distributed $Δ$-Coloring Plays Hide-and-Seek
摘要：
我们证明了几个新的紧分布下界的经典对称破缺图问题。作为一个基本的工具，我们首先提供了一个新的有见地的证明，任何确定性的分布式算法，计算$Δ$ -着色$Δ$ -定期树需要$Ω(\log_Δn)$轮和任何随机算法需要$Ω(\log_Δ\log n)$轮。我们证明了这一结果表明，自然松弛的$Δ$着色问题是一个不动点的轮消除框架。作为第一个应用程序，我们表明，我们的$Δ$着色下界证明直接延伸到arbdefective着色。我们确切地描述了arbdefective着色问题的哪些变体是“容易”的，而哪些是“困难”的。作为第二个应用程序，我们看到我们的主要贡献，我们使用的结构的不动点作为一个积木来证明下限作为一个功能的$Δ$的一大类分布对称性破缺问题。例如，我们得到了一个紧的下限计算的基本问题的$(2,β)$统治集。这是对问题的最佳现有下限的指数改进，这在[FOCS '20]中得到了证明。我们的下限甚至适用于一个更一般的家庭的问题，允许几乎任意组合的自然约束着色问题，方向问题，和独立集的问题，并提供了一个统一的证明已知和新的下限结果，这些类型的问题。作为$Δ$函数的下界也意味着作为$n$函数的下界。我们得到，例如，树上的最大独立集，需要$Ω(\log n / \log \log n)$轮确定性算法，这是紧的。
英文摘要：
We prove several new tight distributed lower bounds for classic symmetry breaking graph problems. As a basic tool, we first provide a new insightful proof that any deterministic distributed algorithm that computes a $Δ$-coloring on $Δ$-regular trees requires $Ω(\log_Δn)$ rounds and any randomized algorithm requires $Ω(\log_Δ\log n)$ rounds. We prove this result by showing that a natural relaxation of the $Δ$-coloring problem is a fixed point in the round elimination framework. As a first application, we show that our $Δ$-coloring lower bound proof directly extends to arbdefective colorings. We exactly characterize which variants of the arbdefective coloring problem are "easy", and which of them instead are "hard". As a second application, which we see as our main contribution, we use the structure of the fixed point as a building block to prove lower bounds as a function of $Δ$ for a large class of distributed symmetry breaking problems. For example, we obtain a tight lower bound for the fundamental problem of computing a $(2,β)$-ruling set. This is an exponential improvement over the best existing lower bound for the problem, which was proven in [FOCS '20]. Our lower bound even applies to a much more general family of problems that allows for almost arbitrary combinations of natural constraints from coloring problems, orientation problems, and independent set problems, and provides a single unified proof for known and new lower bound results for these types of problems. Our lower bounds as a function of $Δ$ also imply lower bounds as a function of $n$. We obtain, for example, that maximal independent set, on trees, requires $Ω(\log n / \log \log n)$ rounds for deterministic algorithms, which is tight.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
