Distributed weak independent sets in hypergraphs: Upper and lower bounds

TL;DR

This paper investigates the problem of finding weak independent sets in hypergraphs within distributed networks, providing new algorithms with upper bounds and establishing lower bounds for these problems.

Contribution

It introduces novel algorithms for computing weak independent sets in hypergraphs, including deterministic and randomized methods, and establishes fundamental lower bounds for these problems.

Findings

Deterministic algorithms with O(Δr / (β - α + 1) + log* n) rounds.

Randomized algorithms with zero rounds under certain conditions.

Lower bounds of Ω(Δ + log* n) and Ω(r + log* n) for specific hypergraph problems.

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.