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

Given a finite set V, and integers k ≥ 1 and r ≥ 0, let us denote by A (k, r) the class of hypergraphs A ⊆ 2V 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 C 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', ∅) 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 e 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.