Maximum shortest path interdiction problem by upgrading edges on trees under weighted $$l_1$$ l 1 norm
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Maximum shortest path interdiction problem by upgrading edges on trees under weighted l1 norm Qiao Zhang1 · Xiucui Guan1
· Panos M. Pardalos2,3
Received: 18 January 2020 / Accepted: 28 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Network interdiction problems by deleting critical edges have wide applicatio ns. However, in some practical applications, the goal of deleting edges is difficult to achieve. We consider the maximum shortest path interdiction problem by upgrading edges on trees (MSPIT) under unit/weighted l1 norm. We aim to maximize the the length of the shortest path from the root to all the leaves by increasing the weights of some edges such that the upgrade cost under unit/weighted l1 norm is upper-bounded by a given value. We construct their mathematical models and prove some properties. We propose a revised algorithm for the problem (MSPIT) under unit l1 norm with time complexity O(n), where n is the number of vertices in the tree. We put forward a primal dual algorithm in O(n 2 ) time to solve the problem (MSPIT) under weighted l1 norm, in which a minimum cost cut is found in each iteration. We also solve the problem to minimize the cost to upgrade edges such that the length of the shortest path is lower bounded by a value and present an O(n 2 ) algorithm. Finally, we perform some numerical experiments to compare the results obtained by these algorithms. Keywords Network interdiction problem · Upgrading critical edges · Shortest path · Weighted l1 norm · Primal dual algorithm · Minimum cost cut
1 Introduction Network interdiction problems by deleting critical edges (denoted by (NIP-DE)) have been studied in recent twenty years. The classical problem (NIP-DE) mainly has two types. One is the K -most-critical-edge problem [1–3,5–7,10,12,13], which aims at making some network performance as poor as possible by deleting at most K edges, and the other one is the critical edge interdiction problem [4,17], which aims to delete as fewer edges as possible to assure some network performance bounded by a constant. The problem (NIP-DE) has been
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Xiucui Guan [email protected]
1
School of Mathematics, Southeast University, Nanjing 210096, China
2
Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA
3
LATNA, Higher School of Economics, Moscow, Russia
123
Journal of Global Optimization
widely studied in some main network performances including a shortest path [1,2,6,11,13], a minimum spanning tree [5,8,10,12,15], a maximum matching [4,16,17,19], a maximum flow [18,20] and a center or median location [3] etc. They have wide applications in communication networks, transportation networks, network war and terrorist networks [7,11]. The problem (NIP-DE) was first applied to the shortest path problem by Corley and Sha in [6]. For any K , Bar-Noy et al. [1] showed that it is N P-hard. Khachiyan et al. [11] showed that there is no approximation algorithm with approximation ratio 2, which i
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