Minimum Transversals in Posi-modular Systems

Given a system (V, f, d) on a finite set V consisting of two set functions f : 2 → R and d : 2 → R, we consider the problem of finding a set R ⊆ V of minimum cardinality such that f(X) ≥ d(X) for all X ⊆ V − R, where the problem can be regarded as a natural generalization of the source location problems and the external network problems in (undirected) graphs and hypergraphs. We give a structural characterization of minimal deficient sets of (V, f, d) under certain conditions. We show that all such sets form a tree hypergraph if f is posi-modular and d is modulotone (i.e., each nonempty subset X of V has an element v ∈ X such that d(Y ) ≥ d(X) for all subsets Y of X that contain v), and that conversely any tree hypergraph can be represented by minimal deficient sets of (V, f, d) for a posi-modular function f and a modulotone function d. By using this characterization, we present a polynomial-time algorithm if, in addition, f is submodular and d is given by either d(X) = max{p(v) | v ∈ X} for a function p : V → R+ or d(X) = max{r(v, w) | v ∈ X, w ∈ V − X} for a function r : V 2 → R+. Our result provides first polynomial-time algorithms for the source location problem in hypergraphs and the external network problems in graphs and hypergraphs. We also show that the problem is intractable, even if f is submodular and d ≡ 0.