On the Complexity of the F-Neighbor P-Center Problem

Abstract

The problem α-Neighbor p-Center is a generalization of the classic problem p-Center. Given an undirected complete network G with edge-distances ful lling the triangle inequality, the goal is to minimize the maximum distance of a client to its α nearest neighbors in the set of p centers. This problem has been proven to be NP-complete for any xed value of α. In this paper, we generalize this problem further by introducing the problem p-Center(f), where instead of a constant α we have a xed function f in p and |V (G)|, i.e. we are considering the maximum distance of a client to its f(p,|V (G)|) nearest neighboring centers. We show that this problem is polynomial-time solvable in the case that f(p,n) = p holds for every p and n, but remains NP-hard for any f where 1 ≤ f(p,n) ≤ p − 1 holds for all p and n. Also, it does not admit a polynomial-time approximation with a ratio better than 2 in this case, unless P = NP holds.

Keywords: p-center problem, location theory, networks