Generating Maximal Independent Sets for Hypergraphs with Bounded Edge-Intersections

Abstract

Given a finite set V, and integers k ≥ 1 and r ≥ 0, denote by $\mathbb{A}(k,r)$ the class of hypergraphs $\mathcal{A}\subseteq 2^{v}$ with (k,r)-bounded intersections, i.e. in which the intersection of any k distinct hyperedges has size at most r. We consider the problem MIS(A,I): given a hypergraph $\mathcal{A}$ and a subfamily $\mathcal{I}\subseteq\mathcal{I}\mathcal{(A)}$ , of its maximal independent sets (MIS) $\mathcal{I}\mathcal{(A)}$ , either extend this subfamily by constructing a new MIS $I \in \mathcal{I}\mathcal{(A)}\backslash \mathcal{I}$ or prove that there are no more MIS, that is $\mathcal{I} = \mathcal{I}\mathcal{(A)}$ . We show that for hypergraphs $\mathcal{A}\in \mathbb{A}(k,r)$ with k + r ≤ const, problem MIS( $\mathcal{A, I}$ ) is NC-reducible to problem MIS(A′, θ) of generating a single MIS for a partial subhypergraph A′ of $\mathcal{A}$ . In particular, for this class of hypergraphs, we get an incremental polynomial algorithm for generating all MIS. Furthermore, combining this result with the currently known algorithms for finding a single maximal independent set of a hypergraph, we obtain efficient parallel algorithms for incrementally generating all MIS for hypergraphs in the classes $\mathbb{A}(1,c)$ , $\mathbb{A}(c,0)$ , and $\mathbb{A}(2,1)$ , where c is a constant. We also show that, for $\mathcal{A}\in \mathbb{A}(k,r)$ , where k + r ≤ const, the problem of generating all MIS of $\mathcal{A}$ can be solved in incremental polynomial-time with space polynomial only in the size of $\mathcal{A}$ .