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 0(Δ^λ(π) T _π(n)(log n)²) using a polynomial number of processors, where λ(π) is the maximum of diameters of minimal graphs violating π and T _π(n) is the time needed to decide whether a graph with n vertices satisfies π. (2) Given a family C = c ₁,...,c _m of subsets of a finite set S = 1,..., n with S = (U ^m_i=1 c_i, a 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{¦c_i,¦ ¦i=1,..., m and Β = max{¦d_j¦ ¦j = 1,..., n.