Complexity of determining the most vital elements for the p-median and p-center location problems

We consider the k most vital edges (nodes) and min edge (node) blocker versions of the p-median and p-center location problems. Given a weighted connected graph with distances on edges and weights on nodes, the k most vital edges (nodes) p-median (respectively p-center) problem consists of finding a subset of k edges (nodes) whose removal from the graph leads to an optimal solution for the p-median (respectively p-center) problem with the largest total weighted distance (respectively maximum weighted distance). The complementary problem, min edge (node) blocker p-median (respectively p-center), consists of removing a subset of edges (nodes) of minimum cardinality such that an optimal solution for the p-median (respectively p-center) problem has a total weighted distance (respectively a maximum weighted distance) at least as large as a specified threshold. We show that k most vital edges p-median and k most vital edges p-center are NP-hard to approximate within a factor and respectively, for any I mu > 0, while k most vital nodes p-median and k most vital nodes p-center are NP-hard to approximate within a factor , for any I mu > 0. We also show that the complementary versions of these four problems are NP-hard to approximate within a factor 1.36.