Minimum Transversals in Posi-modular Systems

Abstract

Given a system (V,f,d) on a finite set V consisting of two set functions f:2V → R and d:2V → 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 → R+ or d(X)=max{r(v,w) |v∈X, w∈V-X} for a function T: V2 → 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.