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 with a nonnegative edge length function ℓ, a vertex set , and an integer p, , let denote the shortest distance from v to C of G for each vertex v in , and the eccentricity of C denote . The connected p-centre problem is to find a vertex set P in V, , such that the eccentricity of P is minimized but the induced subgraph of P must be connected. Given an undirected graph and an integer , 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 . 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 , . 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 , , for the converse connected p-centre problem unless .

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