On Computing Maximal Independent Sets of Hypergraphs in Parallel

Abstract

Whether or not the problem of finding maximal independent sets (MIS) in hypergraphs is in (R)NC is one of the fundamental problems in the theory of parallel computing. Essentially, the challenge is to design (randomized) algorithms in which the number of processors used is polynomial and the (expected) runtime is polylogarithmic in the size of the input. Unlike the well-understood case of MIS in graphs, for the hypergraph problem, our knowledge is quite limited despite considerable work. It is known that the problem is in RNC when the edges of the hypergraph have constant size. For general hypergraphs with n vertices and m edges, the fastest previously known algorithm works in time O (√ n ) with poly( m,n ) processors. In this article, we give an EREW PRAM randomized algorithm that works in time n o (1) with O ( n + mlog n ) processors on general hypergraphs satisfying m ≤ n o (1) log log n /log log log n . We also give an EREW PRAM deterministic algorithm that runs in time n ϵ on a graph with m ≤ n 1/δ edges, for any constants δ, ϵ; the number of processors is polynomial in m, n for a fixed choice of δ, ϵ. Our algorithms are based on a sampling idea that reduces the dimension of the hypergraph and employs the algorithm for constant dimension hypergraphs as a subroutine.