Minimizing a monotone concave function with laminar covering constraints

Mariko Sakashita, Kazuhisa Makino, Satoru Fujishige
2008 Discrete Applied Mathematics  
Let V be a finite set with |V | = n. A family F ⊆ 2 V is called laminar if for all two sets X, Y ∈ F, X ∩ Y = ∅ 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 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(n 2 q) time if F can be decomposed into monotone concave functions by
more » ... he 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 (n 2 q) time to solve the problem, which implies that our O(n 2 q) time algorithm is optimal in this case. Furthermore, we propose an O(n log 2 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 (2 n/2 q) time when F is given implicitly by an oracle, and that it is NP-hard if F is given explicitly in a functional form.
doi:10.1016/j.dam.2007.04.016 fatcat:cxsflulerrdvfamvqvrvhlifcq