极大匹配与极大独立集的下界
Lower bounds for maximal matchings and maximal independent sets
摘要：
在$O(Δ+ \log^* n)$通信轮中，有分布式图算法用于寻找最大匹配和最大独立集;这里$n$是节点数，$Δ$是最大度。Linial（1987，1992）的下限表明对$n$的依赖性是最优的：这些问题不能在$o(\log^* n)$轮中解决，即使$Δ= 2$。然而，对$Δ$的依赖性是一个长期存在的开放问题，目前在上限和下限之间存在指数差距。我们证明了上界是紧的。我们发现，任何算法，找到一个最大的匹配或最大的独立集的概率至少$1-1/n$需要$Ω(\min\{Δ,\log \log n / \log \log \log n\})$轮的分布式计算的并行模型。作为一个推论，任何找到最大匹配或最大独立集的确定性算法都需要$Ω(\min\{Δ, \log n / \log \log n\})$轮;这是对先前下界的改进，也是$n$的函数。
英文摘要：
There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(Δ+ \log^* n)$ communication rounds; here $n$ is the number of nodes and $Δ$ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on $n$ is optimal: these problems cannot be solved in $o(\log^* n)$ rounds even if $Δ= 2$. However, the dependency on $Δ$ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least $1-1/n$ requires $Ω(\min\{Δ,\log \log n / \log \log \log n\})$ rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires $Ω(\min\{Δ, \log n / \log \log n\})$ rounds; this is an improvement over prior lower bounds also as a function of $n$.
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