Minimum Transversals in Posi-modular Systems

Abstract

Given a system (V,f,d) on a finite set V consisting of two set functions $f:2^V\rightarrow{\mathbb R}$ and $d:2^V\rightarrow{\mathbb R}$, we consider the problem of finding a set R ⊆ V of the 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 $p:V \rightarrow{\mathbb R}_+$ or d(X)=max{r(v,w) |v∈X, w∈V–X} for a function $r:V^2\rightarrow{\mathbb R}_+$. 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.