Abstract
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 \(\frac{7}{5}-\epsilon\) and \(\frac{4}{3}-\epsilon\) respectively, for any ϵ>0, while k most vital nodes p-median and k most vital nodes p-center are NP-hard to approximate within a factor \(\frac{3}{2}-\epsilon\), for any ϵ>0. We also show that the complementary versions of these four problems are NP-hard to approximate within a factor 1.36.
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Bazgan, C., Toubaline, S. & Vanderpooten, D. Complexity of determining the most vital elements for the p-median and p-center location problems. J Comb Optim 25, 191–207 (2013). https://doi.org/10.1007/s10878-012-9469-8
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DOI: https://doi.org/10.1007/s10878-012-9469-8