Distributed weak independent sets in hypergraphs: Upper and lower bounds

Abstract: In this paper, we consider the problem of finding weak independent sets in a distributed network represented by a hypergraph. In this setting, each edge contains a set of r vertices rather than simply a pair, as in a standard graph. A k-weak independent set in a hypergraph is a set where no edge contains more than k vertices in the independent set. We focus two variations of this problem. First, we study the problem of finding k-weak maximal independent sets, k-weak independent sets where each vertex belongs to at least one edge with k vertices in the independent set. Second we introduce a weaker variant that we call (\alpha, \beta)-independent sets where the independent set is \beta-weak, and each vertex belongs to at least one edge with at least \alpha vertices in the independent set. Finally, we consider the problem of finding a (2, k)-ruling set on hypergraphs, i.e. independent sets where no vertex is a distance of more than k from the nearest member of the set. Given a hypergraph H of rank r and maximum degree \Delta, we provide a LLL formulation for finding an (\alpha, \beta)-independent set when (\beta - \alpha)^2 / (\beta + \alpha) \geq 6 \log(16 r \Delta), an O(\Delta r / (\beta - \alpha + 1) + \log^* n) round deterministic algorithm finding an (\alpha, \beta)-independent set, and a O(\Delta^2(r - k) \log r + \Delta \log r \log^* r + \log^* n) round algorithm for finding a k-weak maximal independent set. Additionally, we provide zero round randomized algorithms for finding (\alpha, \beta) independent sets, when (\beta - \alpha)^2 / (\beta + \alpha) \geq 6 c \log n + 6 for some constant c, and finding an m-weak independent set for some m \geq r / 2k where k is a given parameter. Finally, we provide lower bounds of \Omega(\Delta + \log^* n) and \Omega(r + \log^* n) on the problems of finding a k-weak maximal independent sets for some values of k.