Minimizing a monotone concave function with laminar covering constraints

• Authors: Mariko Sakashita;Kazuhisa Makino;Satoru Fujishige
• Venue: ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
• Year: 2005

Let V be a finite set with |V|=n. A family $\mathcal{F}\subseteq 2^{V}$ is called laminar if for arbitrary two sets $X, Y \in \mathcal{F}$, X ∩ Y ≠ ∅ implies X⊆Y or X⊇Y. Given a laminar family $\mathcal{F}$, a demand function d →ℝ+, and a monotone concave cost function $F : \mathbb{R}_{+}^{V} \rightarrow \mathbb{R}_{+}$, we consider the problem of finding a minimum-cost $x \in \mathbb{R}_{+}^{V}$ such that x(X)≥ d(X) for all $X \in \mathcal{F}$. Here we do not assume that the cost function F is differentiable 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 $\mathcal{F}$, where q is the time required for the computation of F(x) for any $x \in \mathbb{R}_{+}^{V}$. We also prove that if F is given by an oracle, then it takes ${\it \Omega}(n^{2}q)$ time to solve the problem, which implies that our O(n2q) time algorithm is optimal in this case. Furthermore, we propose an O(n log2n) 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 ${\it \Omega}(2^{n \over 2}q)$ time when F is given implicitly by an oracle, and that it is NP-hard if F is given explicitly.