Using maximal independent sets to solve problems in parallel

Abstract

By using an O((log n)2) time EREW PRAM algorithm for a maximal independent set problem (MIS), we show the following two results: (1) Given a graph, the maximal vertex-induced subgraph satisfying a hereditary graph property π can be found in time O(δ λ(π) Tπ(n)(log n)2) using a polynomial number of processors, where λ(π) is the maximum of diameters of minimal graphs violating π and Tπ(n) is the time needed to decide whether a graph with n vertices satisfies π. (2) Given a family C = {c1,….,cm} of subsets of a finite set S = {1,…,n} with S = Umi=1ci minimal set cover for S can be computed on an EREW PRAM in time O(αβ(log(n + m))2) using a polynomial number of processors, where α = max{|ci| i = 1,…, m} and β = max{|dj|| j=1,…,n}.