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 $p_r: V \times 2^V \rightarrow \mathbb{R}$ 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 ${\cal W}$ for the transversal problem for π-monotone function r, i.e., there exists a basic tree T for ${\cal W}$ 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.