Interdicting facilities in tree networks

Abstract

This article investigates a network interdiction problem on a tree network: given a subset of nodes chosen as facilities, an interdictor may dissect the network by removing a size-constrained set of edges, striving to worsen the established facilities best possible. Here, we consider a reachability objective function, which is closely related to the covering objective function: the interdictor aims to minimize the number of customers that are still connected to any facility after interdiction. For the covering objective on general graphs, this problem is known to be NP-complete (Fröhlich and Ruzika In: On the hardness of covering-interdiction problems. Theor. Comput. Sci., 2021). In contrast to this, we propose a polynomial-time solution algorithm to solve the problem on trees. The algorithm is based on dynamic programming and reveals the relation of this location-interdiction problem to knapsack-type problems. However, the input data for the dynamic program must be elaborately generated and relies on the theoretical results presented in this article. As a result, trees are the first known graph class that admits a polynomial-time algorithm for edge interdiction problems in the context of facility location planning.

p-median location interdiction on trees

“…The first line (11) again follows from inequality (4), ( 12) is due to the definition of q 2 , (13) follows from condition (2) and the last estimation ( 14) follows from inequality (7).…”

“…Here, the authors combine the covering problem with an edge interdiction problem. In [11], the authors present a polynomial time algorithm for the interdiction problem on trees, where an upfront chosen set of facilities is given and the interdictor wishes to worsen the reachability within the tree. In [9], the authors combine the median location problem with edge interdiction.…”