Posi-modular systems with modulotone requirements under permutation constraints

Abstract

Given a system (V,f,r) on a finite set V consisting of a posi-modular function f: 2 V →ℝ and a modulotone function r: 2 V →ℝ, we consider the problem of finding a minimum set R ⊂ V such that f(X)≥r(X) for all X ⊂ V-R. The problem, called the transversal problem, was introduced by Sakashita et al. [6] as a natural generalization of the source location problem and external network problem with edge-connectivity requirements in undirected graphs and hypergraphs. By generalizing [8] for the source location problem, we show that the transversal problem can be solved by a simple greedy algorithm if r is π-monotone, where a modulotone function r is π-monotone if there exists a permutation π of V such that the function associated with r satisfies p r (u,W)≥p r (v, W) for all W⊂V and u,v V with π(u)≥π(v). Here we show that any modulotone function r can be characterized by p r as r(X)= max {p r (v,W)|v X⊂V-W}. We also show the structural properties on the minimal deficient sets for the transversal problem for π-monotone function r, i.e., there exists a basic tree T for such that π(u)≤π(v) for all arcs (u,v) in T, which, as a corollary, gives an alternative proof for the correctness of the greedy algorithm for the source location problem. Furthermore, we show that a fractional version of the transversal problem can be solved by the algorithm similar to the one for the transversal problem. © 2009 Springer-Verlag Berlin Heidelberg.