树中MIS和有界（外）度控制集的改进分布下界
Improved Distributed Lower Bounds for MIS and Bounded (Out-)Degree Dominating Sets in Trees
摘要：
最近，Balliu，Brandt和Olivetti [FOCS '20]给出了树中最大独立集（MIS）问题的第一个$ω(\log^* n)$下界。在这项工作中，我们证明了一个更宽松的家庭分布对称性破缺问题的下限。作为一个副产品，我们得到了改进的下界的分布式MIS问题的树。对于一个参数$k$和一个图$G$的边的方向，如果$S$是$G$的一个控制集，并且在导出子图$G[S]$中，$S$中的每个节点最多有$k$的出度，我们称$G$的节点的子集$S$是$k$ -出度控制集.请注意，对于$k=0$，此定义与MIS的定义一致。对于给定的$k$，我们考虑计算一个$k$ -出度控制集的问题.我们表明，即使在最$Δ$度的正则树中，在标准\LOCAL模型中，也存在一个常数$ε>0$，使得对于$k\leq Δ^ε$，对于计算$k$ -outdegree支配集的问题，任何随机算法至少需要$Ω(\min\{\logΔ,\sqrt{\log\log n}\})$轮，任何确定性算法至少需要$Ω(\min\{\logΔ,\sqrt{\log n}\})$轮。我们的下界的证明是基于最近非常成功的轮消除技术。我们提供了一种新的方法来做轮消除，我们希望是独立的利益简化。我们的新证明比[FOCS '20]中的下界证明简单得多。特别是，我们的轮消证明使用了一系列只能用恒定数量的标签来描述的问题。[FOCS '20]的作者认为MIS问题的这种证明的存在是不可能的。
英文摘要：
Recently, Balliu, Brandt, and Olivetti [FOCS '20] showed the first $ω(\log^* n)$ lower bound for the maximal independent set (MIS) problem in trees. In this work we prove lower bounds for a much more relaxed family of distributed symmetry breaking problems. As a by-product, we obtain improved lower bounds for the distributed MIS problem in trees. For a parameter $k$ and an orientation of the edges of a graph $G$, we say that a subset $S$ of the nodes of $G$ is a $k$-outdegree dominating set if $S$ is a dominating set of $G$ and if in the induced subgraph $G[S]$, every node in $S$ has outdegree at most $k$. Note that for $k=0$, this definition coincides with the definition of an MIS. For a given $k$, we consider the problem of computing a $k$-outdegree dominating set. We show that, even in regular trees of degree at most $Δ$, in the standard \LOCAL model, there exists a constant $ε>0$ such that for $k\leq Δ^ε$, for the problem of computing a $k$-outdegree dominating set, any randomized algorithm requires at least $Ω(\min\{\logΔ,\sqrt{\log\log n}\})$ rounds and any deterministic algorithm requires at least $Ω(\min\{\logΔ,\sqrt{\log n}\})$ rounds. The proof of our lower bounds is based on the recently highly successful round elimination technique. We provide a novel way to do simplifications for round elimination, which we expect to be of independent interest. Our new proof is considerably simpler than the lower bound proof in [FOCS '20]. In particular, our round elimination proof uses a family of problems that can be described by only a constant number of labels. The existence of such a proof for the MIS problem was believed impossible by the authors of [FOCS '20].
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