Approximability results for the converse connected p-centre problem

In this paper, we investigate a combinatorial optimization problem, called the converse connected p-centre problem which is the converse problem of the connected p-centre problem. This problem is a variant of the p-centre problem. Given an undirected graph G = (V, E, l) with a nonnegative edge length function l, a vertex set C subset of V, and an integer p, 0 < p < vertical bar V vertical bar, let d(v, C) denote the shortest distance from v to C of G for each vertex v in V \ C, and the eccentricity ecc(C) of C denote max(v is an element of V) d(v, C). The connected p-centre problem is to find a vertex set P in V, vertical bar P vertical bar = p, such that the eccentricity of P is minimized but the induced subgraph of P must be connected. Given an undirected graph G = (V, E, l) and an integer gamma > 0, the converse connected p-centre problem is to find a vertex set P in V with minimum cardinality such that the induced subgraph of P must be connected and the eccentricity ecc(P) <= gamma. One of the applications of the converse connected p-centre problem has the facility location with load balancing and backup constraints. The connected p-centre problem had been shown to be NP-hard. However, it is still unclear whether there exists a polynomial time approximation algorithm for the converse connected p-centre problem. In this paper, we design the first approximation algorithm for the converse connected p-centre problem with approximation ratio of (1 + epsilon) ln vertical bar V vertical bar, epsilon > 0. We also discuss the approximation complexity for the converse connected p-centre problem. We show that there is no polynomial time approximation algorithm achieving an approximation ratio of (1 - epsilon) ln vertical bar V vertical bar, epsilon > 0, for the converse connected p-centre problem unless P = NP.