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

Bazgan, Cristina; Toubaline, Sonia; Vanderpooten, Daniel

doi:10.1007/s10878-012-9469-8

Cited by 10 publications

(6 citation statements)

References 12 publications

(19 reference statements)

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“…Proof. Let r ∈ R N ≥0 and consider the maximization problem that defines ψ(r), which is the same as the linear program desribed by (5). Furthermore, let r ′ = r ∧ 1, and let U = {e ∈ N | r(e) ≥ 1}.…”

Section: General Approach To Obtain 2-pseudoapproximationsmentioning

confidence: 99%

“…A significant effort has been dedicated to understanding interdiction problems. The list of optimization problems for which interdiction variants have been studied includes maximum flow [31,32,26,34], minimum spanning tree [12,36], shortest path [3,20], connectivity of a graph [35], matching [33,25], matroid rank [17,18], stable set [4], several variants of facility location [9,5], and more.…”

Section: Introductionmentioning

confidence: 99%

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Interdicting Structured Combinatorial Optimization Problems with {0, 1}-Objectives

Chestnut¹,

Zenklusen²

2017

Mathematics of OR

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Interdiction problems ask about the worst-case impact of a limited change to an underlying optimization problem. They are a natural way to measure the robustness of a system, or to identify its weakest spots. Interdiction problems have been studied for a wide variety of classical combinatorial optimization problems, including maximum s-t flows, shortest s-t paths, maximum weight matchings, minimum spanning trees, maximum stable sets, and graph connectivity. Most interdiction problems are NPhard, and furthermore, even designing efficient approximation algorithms that allow for estimating the order of magnitude of a worst-case impact, has turned out to be very difficult. Not very surprisingly, the few known approximation algorithms are heavily tailored for specific problems.Inspired by an approach of Burch et al.[8], we suggest a general method to obtain pseudoapproximations for many interdiction problems. More precisely, for any α > 0, our algorithm will return either a (1 + α)-approximation, or a solution that may overrun the interdiction budget by a factor of at most 1 + α −1 but is also at least as good as the optimal solution that respects the budget. Furthermore, our approach can handle submodular interdiction costs when the underlying problem is to find a maximum weight independent set in a matroid, as for example the maximum weight forest problem. Additionally, our approach can sometimes be refined by exploiting additional structural properties of the underlying optimization problem to obtain stronger results. We demonstrate this by presenting a PTAS for interdicting b-stable sets in bipartite graphs.

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Section: General Approach To Obtain 2-pseudoapproximationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Interdicting Structured Combinatorial Optimization Problems with {0, 1}-Objectives

Chestnut¹,

Zenklusen²

2017

Mathematics of OR

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“…They show that in general graphs, the problem of the follower is NP-complete, whereas the problem of the leader is even Σ p 2 -complete. In Bazgan et al (2010Bazgan et al ( , 2013, inapproximability results for location-interdiction problems with weighted center and median objectives are presented.…”

Section: Introductionmentioning

confidence: 99%

Interdicting facilities in tree networks

Fröhlich

Ruzika

2021

TOP

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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.

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“…It is often easy to prove NP‐hardness for interdiction variants of combinatorial optimization problems, but there are only a handful of approximation hardness theorems for interdiction problems. We have already mentioned that matroid rank interdiction is D

k

S‐hard, other known lower bounds include APX hardness of shortest path interdiction by removal of edges ,

k

‐median and

k

‐center interdiction , and assignment problem interdiction as well as clique hardness for spanning tree interdiction by removal of vertices . Approximation or bicriteria approximation algorithms are known for the interdiction versions of minimum spanning tree by removal of edges , (fractional) multicommodity flow , matching , and more generally packing LPs .…”

Section: Introductionmentioning

confidence: 99%

Hardness and approximation for network flow interdiction

Chestnut

Zenklusen

2017

Networks

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In the Network Flow Interdiction problem, an adversary attacks a network in order to minimize the maximum s‐t‐flow. Very little is known about the approximatibility of this problem, despite decades of interest in it. It is, surprisingly, nontrivial to obtain in polynomial time an approximation guarantee that is independent of the number of edges in the graph and their capacities. We present the first such approximation algorithm, which has approximation ratio at most 2(n−1) for any graph with n vertices. We complement the algorithm with a hardness theorem. Past work has shown that Network Flow Interdiction cannot be much easier to approximate than Densest k‐Subgraph. We show that any nϵ‐approximation algorithm for Network Flow Interdiction, or one of several variants, would imply a 2n4ϵ‐approximation algorithm for Densest k‐Subgraph, which is an improvement over past work in terms of the polynomial factor. We also show that Network Flow Interdiction is essentially the same as the Budgeted Minimum s‐t‐Cut problem, and transferring our results gives hardness and an approximation algorithm for that problem, as well. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 378–387 2017

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

Cited by 10 publications

References 12 publications

Interdicting Structured Combinatorial Optimization Problems with {0, 1}-Objectives

Interdicting Structured Combinatorial Optimization Problems with {0, 1}-Objectives

Interdicting facilities in tree networks

Hardness and approximation for network flow interdiction

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