An efficient implementation of a quasi-polynomial algorithm for generating hypergraph transversals and its application in joint generation

Given a finite set V , and a hypergraph H ⊆ 2 , the hypergraph transversal problem calls for enumerating all minimal hitting sets (transversals) for H. This problem plays an important role in practical applications as many other problems were shown to be polynomially equivalent to it. Fredman and Khachiyan (1996) gave an incremental quasi-polynomial time algorithm for solving the hypergraph transversal problem. In this paper, we present an efficient implementation of this algorithm. While we show that our implementation achieves the same theoretical worst case bound, practical experience with this implementation shows that it can be substantially faster. We also show that a slight modification of the original algorithm can be used to give a stronger bound on the running time. More generally, we consider a monotone property π over a bounded n-dimensional integral box. As an important application of the above hypergraph transversal problem, pioneered by Bioch and Ibaraki (1995), we consider the problems of incrementally generating simultaneously both families of all minimal subsets satisfying π and all maximal subsets not satisfying π, for properties given by a polynomial-time satisfiability oracle. Problems of this type arise in many practical applications. It is known that the above joint generation problem can be solved in incremental quasi-polynomial time performing a polynomialtime reduction to a generalization of the hypergraph transversal problem on integer boxes. In this paper we present an efficient implementation of this procedure, and present experimental results to evaluate our implementation for a number of interesting monotone properties π.