We investigate the distributed complexity of maximal matching and maximal independent set (MIS) in hypergraphs in the LOCAL model. A maximal matching of a hypergraph $H=(V_H,E_H)$ is a maximal disjoint set $M\subseteq E_H$ of hyperedges and an MIS $S\subseteq V_H$ is a maximal set of nodes such that no hyperedge is fully contained in $S$. Both problems can be solved by a simple sequential greedy algorithm, which can be implemented naively in $O(\Delta r + \log^* n)$ rounds, where $\Delta$ is the maximum degree, $r$ is the rank, and $n$ is the number of nodes. We show that for maximal matching, this naive algorithm is optimal in the following sense. Any deterministic algorithm for solving the problem requires $\Omega(\min\{\Delta r, \log_{\Delta r} n\})$ rounds, and any randomized one requires $\Omega(\min\{\Delta r, \log_{\Delta r} \log n\})$ rounds. Hence, for any algorithm with a complexity of the form $O(f(\Delta, r) + g(n))$, we have $f(\Delta, r) \in \Omega(\Delta r)$ if $g(n)$ is not too large, and in particular if $g(n) = \log^* n$ (which is the optimal asymptotic dependency on $n$ due to Linial's lower bound [FOCS'87]). Our lower bound proof is based on the round elimination framework, and its structure is inspired by a new round elimination fixed point that we give for the $\Delta$-vertex coloring problem in hypergraphs. For the MIS problem on hypergraphs, we show that for $\Delta\ll r$, there are significant improvements over the naive $O(\Delta r + \log^* n)$-round algorithm. We give two deterministic algorithms for the problem. We show that a hypergraph MIS can be computed in $O(\Delta^2\cdot\log r + \Delta\cdot\log r\cdot \log^* r + \log^* n)$ rounds. We further show that at the cost of a worse dependency on $\Delta$, the dependency on $r$ can be removed almost entirely, by giving an algorithm with complexity $\Delta^{O(\Delta)}\cdot\log^* r + O(\log^* n)$.
翻译:我们调查了最大匹配和最高独立设置( MIS) 的分布复杂度, 在LOCAL 模型中, 最高匹配度是最高不连接值$H=( V_ H, E_ H) 最高匹配度, 最高不连接值$M\ subseteq E_ H$, 最高不匹配值$S\ subseteq V_ H$( 最高不包含最高不包含在$S$中 。 两种问题都可以通过简单的顺序贪婪算法来解决。 两种问题都可以通过简单的顺序调算法解决, 可以用美元( delta r+\ log_ n美元 美元) 来天真切地执行, 美元是最高不比值$D$, 美元比值是最低的。