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

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 edgeconnectivity 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.