Minimizing a monotone concave function with laminar covering constraints

Abstract

Let V be a finite set with |V| = n. A family F ⊆ 2V is called laminar if for arbitrary two sets X, Y ∈ F, X∩Y ≠ θ implies X ⊆ Y or X ⊇ Y. Given a laminar family F, a demand function d : F → ℝ+, and a monotone concave cost function F : ℝ+V → ℝ+, we consider the problem of finding a minimum-cost x ∈ ℝ+Vsuch that x(X) ≥ d(X) for all X ∈ F. Here we do not assume that the cost function F is differentiate or even continuous. We show that the problem can be solved in O(n2q) time if F can be decomposed into monotone concave functions by the partition of V that is induced by the laminar family F, where q is the time required for the computation of F(x) for any x ∈ ℝ +V. We also prove that if F is given by an oracle, then it takes Ω(n2q) time to solve the problem, which implies that our O(n2q) time algorithm is optimal in this case. Furthermore, we propose an O(n log2 n) algorithm if F is the sum of linear cost functions with fixed setup costs. These also make improvements in complexity results for source location and edge-connectivity augmentation problems in undirected networks. Finally, we show that in general our problem requires Ω(2n/2q) time when F is given implicitly by an oracle, and that it is NP-hard if F is given explicitly. © Springer-Verlag Berlin Heidelberg 2005.