Worst-Case Incremental Analysis for a Class of p-Facility Location Problems

We consider a rather large class of p-facility location models including the p-median, p-center, and other related and more general models. For any such model of interest with p new facilities, let v(p) denote the minimal objective function value and let n be the number of demand points. Given 1 ≤ p < q ≤ n, we find easily computed positive constants k(p, q), where v(q)/v(p) ≤ k(p, q) ≤ 1. These resulting inequalities relating v(p) and v(q) are worst case, since they are attained as equalities for a class of "hub-and-spoke" trees. Our results also provide insight into some demand point aggregation problems, where a graph of the function v(q) can provide an upper bound on aggregation error.