Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Parameterized Complexity
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
The root location problem for arc-disjoint arborescences
Discrete Applied Mathematics
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Let G = (V, E) be an undirected multi-graph where V and E are a set of nodes and a set of edges, respectively. Let k and l be fixed nonnegative integers. This paper considers location problems of finding a minimum size of node-subset S ⊆ V such that node-connectivity between S and x is greater than or equal to k and edge-connectivity between S and x is greater than or equal to l for every x ∈ V . This problem has important applications for multi-media network control and design. For a problem of considering only edge-connectivity, i.e., k = 0, an O(L(|V|, |E|, l)) = O(|E| + |V|2 + |V|min{|E|, l|V|}min{l, |V|}) time algorithm was already known, where L(|V|, |E|, l) is a time to find all h-edge-connected components for h = 1, 2, ..., l. This paper presents an O(L(|V|, |E|, l)) time algorithm for 0 ≤ k ≤ 2 and l ≥ 0. It also shows that if k ≥ 3, the problem is NP-hard even for l = 0. Moreover, it shows that if the size of S is regarded as a parameter, the parameterized problem for k = 3 and l ≤ 1 is FPT (fixed parameter tractable).