Minimum Transversals in Posimodular Systems

Abstract

Given a system $(V,f,d)$ on a finite set V consisting of two set functions $f:2^V\to\mathbb{R}$ and $d:2^V\to\mathbb{R}$, we consider the problem of finding a set $R\subseteq V$ of minimum cardinality such that $f(X)\ge d(X)$ for all $X\subseteq V-R$, 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 posimodular and d is modulotone (i.e., each nonempty subset X of V has an element $v\in X$ such that $d(Y)\ge d(X)$ 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 posimodular 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 $d(X)=\max\{p(v)\mid v\in X\}$ for a function $p:V\to\RR_+$ or $d(X)=\max\{r(v,w)\mid v\in X,w\in V-X\}$ for a function $r:V^2\to\mathbb{R}_+$. 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 $d\equiv\mathbf{0}$.