A faster algorithm for finding the minimum cut in a directed graph
SODA selected papers from the third annual ACM-SIAM symposium on Discrete algorithms
Graph classes: a survey
A combinatorial algorithm minimizing submodular functions in strongly polynomial time
Journal of Combinatorial Theory Series B
A combinatorial strongly polynomial algorithm for minimizing submodular functions
Journal of the ACM (JACM)
Locating sources to meet flow demands in undirected networks
Journal of Algorithms
The external network problem with edge- or arc-connectivity requirements
CAAN'04 Proceedings of the First international conference on Combinatorial and Algorithmic Aspects of Networking
An algorithm for source location in directed graphs
Operations Research Letters
Combinatorial optimization in system configuration design
Automation and Remote Control
ACM Transactions on Algorithms (TALG)
Augmenting edge-connectivity between vertex subsets
CATS '09 Proceedings of the Fifteenth Australasian Symposium on Computing: The Australasian Theory - Volume 94
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Given a system (V,f,d) on a finite set V consisting of two set functions f:2V → R and d:2V → R, we consider the problem of finding a set R⊆V of the minimum cardinality such that f(X)≥d(X) for all X ⊆ V - R, where the problem can be regarded as a natural generalization of the source location problems and the external network problems in (undirected) graphs and hypergraphs. We give a structural characterization of minimal deficient sets of (V,f,d) under certain conditions. We show that all such sets form a tree hypergraph if f is posi-modular and d is modulotone (i.e., each nonempty subset X of V has an element v∈X such that d(Y)≥d(X) for all subsets Y of X that contain v), and that conversely any tree hypergraph can be represented by minimal deficient sets of (V,f,d) for a posi-modular function f and a modulotone function d. By using this characterization, we present a polynomial-time algorithm if, in addition, f is submodular and d is given by either d(X)=max{p(v)|v∈X} for a function p: V → R+ or d(X)=max{r(v,w) |v∈X, w∈V-X} for a function T: V2 → R+. Our result provides first polynomial-time algorithms for the source location problem in hypergraphs and the external network problems in graphs and hypergraphs. We also show that the problem is intractable, even if f is submodular and d=0.