Posi-modular Systems with Modulotone Requirements under Permutation Constraints

• Authors: Toshimasa Ishii;Kazuhisa Makino
• Venue: ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
• Year: 2009

Given a system (V,f,r) on a finite set V consisting of a posi-modular function f: 2 V 驴驴 and a modulotone function r: 2 V 驴驴, we consider the problem of finding a minimum set R 驴 V such that f(X) 驴 r(X) for all X 驴 V 驴 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 驴-monotone, where a modulotone function r is 驴-monotone if there exists a permutation 驴 of V such that the function $p_r: V \times 2^V \rightarrow \mathbb{R}$ associated with r satisfies p r (u,W) 驴 p r (v, W) for all W 驴 V and u,v 驴 V with 驴(u) 驴 驴(v). Here we show that any modulotone function r can be characterized by p r as r(X) = max {p r (v,W)|v 驴 X 驴 V 驴 W}.We also show the structural properties on the minimal deficient sets ${\cal W}$ for the transversal problem for 驴-monotone function r, i.e., there exists a basic tree T for ${\cal W}$ such that 驴(u) ≤ 驴(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.