On the dualization of hypergraphs with bounded edge-intersections and other related classes of hypergraphs

Abstract

Given a finite set V , and integers k ≥ 1 and r ≥ 0 , let us denote by A ( k , r ) the class of hypergraphs A ⊆ 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 A , and a subfamily I ⊆ I ( A ) of its maximal independent sets (MIS) I ( A ) , either extend this subfamily by constructing a new MIS I ∈ I ( A ) ∖ I or prove that there are no more MIS, that is I = I ( A ) . It is known that, for hypergraphs of bounded dimension A ( 1 , δ ) , as well as for hypergraphs of bounded degree A ( δ , 0 ) (where δ is a constant), problem MIS ( A , I ) can be solved in incremental polynomial time. In this paper, we extend this result to any integers k , r such that k + r = δ is a constant. More precisely, we show that for hypergraphs A ∈ A ( k , r ) with k + r ≤ const , problem MIS ( A , I ) is NC-reducible to the problem MIS ( A ′ , 0̸ ) of generating a single MIS for a partial subhypergraph A ′ of A . In particular, this implies that MIS ( A , I ) is polynomial, and we get an incremental polynomial algorithm for generating all MIS. Furthermore, combining this result with the currently known algorithms for finding a single maximally independent set of a hypergraph, we obtain efficient parallel algorithms for incrementally generating all MIS for hypergraphs in the classes A ( 1 , δ ) , A ( δ , 0 ) , and A ( 2 , 1 ) , where δ is a constant. We also show that, for A ∈ A ( k , r ) , where k + r ≤ const , the problem of generating all MIS of A can be solved in incremental polynomial-time and with space polynomial only in the size of A .