Interdiction problems on planar graphs

Pan, Feng; Schild, Aaron

doi:10.1016/j.dam.2015.05.036

Cited by 20 publications

(14 citation statements)

References 18 publications

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“…This claim appears theoretically significant in understanding the (in)approximability of Problem 2, but may be less practical in handling Problem 2, because, so far, we have not found an approximation algorithm with constant approximation factors for the minimal cost 1‐blocker problem in the literature. To the best of our knowledge, there are pseudo‐polynomial algorithms for solving the minimal cost 1‐blocker problem respectively on graphs with a bounded treewidth proposed by Zenkusen and on planar graphs proposed by Pan and Schild …”

Section: Minimum Cost Link Deletion Problemmentioning

confidence: 99%

Minimal structural perturbations for controllability of a networked system: Complexities and approximations

Zhang

Zhou

2019

Intl J Robust & Nonlinear

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Summary Link additions/deletions and actuator/sensor failures are common structural perturbations for real networked systems. In this paper, we consider three related problems on determining the minimal cost structural perturbations, including link additions, link deletions, and input deletions to make a networked system structurally controllable/uncontrollable, mainly focusing on their computational complexities and approximations. Formally, given a structured system, it is proven that (i) it is NP‐hard to add the minimal cost of links, including links among state variables (ie, state links) and links from the existing inputs to state variables (ie, input links), from a given set of links to make the system structurally controllable; (ii) it is NP‐hard to determine the minimal cost of links whose deletions deteriorate structural controllability of the system, even when the removable links are restricted in either the input links or the state links. It is also proven that determining the minimal cost of inputs whose deletions cause structural uncontrollability is strongly NP‐hard even with dedicated input structure. Furthermore, some fundamental approximation results for these problems are established. These results may serve an answer to the general hardness and approximability of optimally designing (modifying) a structurally controllable network topology and of measuring controllability robustness against link/input failures. Additionally, several polynomial‐time tractable cases of the aforementioned problems are also identified.

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Section: Minimum Cost Link Deletion Problemmentioning

confidence: 99%

Minimal structural perturbations for controllability of a networked system: Complexities and approximations

Zhang

Zhou

2019

Intl J Robust & Nonlinear

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“…Anyhow, there are only few interdiction problems for which approximation algorithms are known, and the gap between these approximations and the best known bound is evident (cf. Zenklusen 2016, 2017;Pan and Schild 2016;Phillips 1993;Zenklusen 2010Zenklusen , 2014. This motivates studying the threshold between exact solvability and hardness in more detail.…”

Section: Introductionmentioning

confidence: 99%

Interdicting facilities in tree networks

Fröhlich

Ruzika

2021

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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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“…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%

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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Interdiction problems on planar graphs

Cited by 20 publications

References 18 publications

Minimal structural perturbations for controllability of a networked system: Complexities and approximations

Minimal structural perturbations for controllability of a networked system: Complexities and approximations

Interdicting facilities in tree networks

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

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