Source location problems considering vertex-connectivity and edge-connectivity simultaneously

• Published in: Networks Vol. 40; no. 2; pp. 63 - 70
• Main Authors: Ito, Hiro, Ito, Motoyasu, Itatsu, Yuichiro, Nakai, Kazuhiro, Uehara, Hideyuki, Yokoyama, Mitsuo
• Format: Journal Article
• Language: English
• Published: New York Wiley Subscription Services, Inc., A Wiley Company 01-09-2002; John Wiley & Sons
• Subjects: Applied sciences; connectivity; Exact sciences and technology; fixed-parameter tractable; Flows in networks. Combinatorial problems; graph; Interconnected networks; location problem; Networks and services in france and abroad; Operational research and scientific management; Operational research. Management science; source; Telecommunications; Telecommunications and information theory; Teleprocessing networks. Isdn; Necessary and sufficient condition; Graph connectivity; NP hard problem; Multigraph; Location problem; Non directed graph; Polynomial time
• ISSN: 0028-3045; 1097-0037
• DOI: 10.1002/net.10034

Let G = (V, E) be an undirected multigraph, where V and E are a set of vertices and a set of edges, respectively. Let k and l be fixed nonnegative integers. This paper considers location problems of finding a minimum‐size vertex‐subset S ⊆ V such that for each vertex x ∈ V the vertex‐connectivity between S and x is greater than or equal to k and the edge‐connectivity between S and x is greater than or equal to l. For the problem with edge‐connectivity requirements, that is, k = 0, an O(L(|V|, |E|, l)) time algorithm is already known, where L(|V|, |E|, l) is the time to find all h‐edge‐connected components for h = 1, 2, … , l and O(L(|V|, |E|, l)) = O(|E| + |V|2 + |V|min{|E|, l|V|}min{l, |V|}) is known. In this paper, we show that the problem with k ≥ 3 is NP‐hard even for l = 0. We then present an O(L(|V|, |E|, l)) time algorithm for 0 ≤ k ≤ 2 and l ≥ 0. Moreover, we prove that the problem parameterized by the size of S is fixed‐parameter tractable (FPT) for k = 3 and l ≥ 0. © 2002 Wiley Periodicals, Inc.