In this paper, we consider source location problems and their generalizations with three connectivity requirements (arc-connectivity requirements λ and two kinds of vertex-connectivity requirements κ andκ), where the source location problems are to find a minimum-cost set S ⊆ V in a given graph G = (V , A) with a capacity function u : A → R + such that for each vertex v ∈ V , the connectivity from S to v (resp., from v to S) is at least a given demand d − (v) (resp., d + (v)). We show that the source location problem with edge-connectivity requirements in undirected networks is strongly NP-hard, which solves an open problem posed by Arata et al. (J. Algorithms 42: 54-68, 2002). Moreover, we show that the source location problems with three connectivity requirements are inapproximable within a ratio of c ln D for some constant c, unless every problem in NP has an O(N log log N )-time deterministic algorithm. Here D denotes the sum of given demands. We also devise (1 + ln D)-approximation algorithms for all the extended source location problems if we have the integral capacity and demand functions. By the inapproximable results above, this implies that all the source location problems are (ln v∈V (d + (v) + d − (v)))-approximable.
Given a system (V, f, d) on a finite set V consisting of two set functions f : 2 V → R and d : 2 V → R, we consider the problem of finding a set R ⊆ V of 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 r : V 2 → 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.
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