Transversal Hypergraphs and Families of Polyhedral Cones

We discuss the complexity of generating certain geometric configurations related to the classical theorems of Caratheodory and Helly. Given a set K of rational cones in n dimensions and a rational n-vector b, we consider the families MI N(K, b) and MAX (K, b) of all minimal and all maximal subsets of K that generate or avoid b, respectively. We show that the problems of generating MIN(K, b) and MAX(K, b) are both NP-hard already for systems of dihedral cones, while the case where all cones in K are rays remains open and includes the well-known problem of enumerating the vertices of a given polytope. On the other hand, for any set of rational cones, the generation of MIN(K, b) ∪ MAX (K, b) is polynomially equivalent to the hypergraph transversal problem, and hence can be carried out in a quasi-polynomial time. We provide other examples of mutually transversal NP-hard families of cones and polytopes.

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