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 : 2V → ℝ and a modulotone function r : 2V → 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 in [M. Sakashita, K. Makino, H. Nagamochi and S. Fujishige, Minimum transversals in posi-modular systems, SIAM J. Discrete Math.23 (2009) 858–871] as a natural generalization of the source location problem and external network problem with edge-connectivity requirements in undirected graphs and hypergraphs. By generalizing [H. Tamura, H. Sugawara, M. Sengoku and S. Shinoda, Plural cover problem on undirected flow networks, IEICE Trans.J81-A (1998) 863–869] 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 pr: V × 2V → ℝ associated with r satisfies pr(u, W) ≥ pr(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 pr as r(X) = max {pr(v, W) | v ∈ X ⊆ V - W}. We also show the structural properties on the minimal deficient sets [Formula: see text] for the transversal problem for π-monotone function r, i.e., there exists a basic tree T for [Formula: see text] 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.