Network interdiction via a Critical Disruption Path: Branch-and-Price algorithms

Granata, Donatella; Steeger, Gregory; Rebennack, Steffen

doi:10.1016/j.cor.2013.04.016

Cited by 22 publications

(17 citation statements)

References 16 publications

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“…In many of these studies, a wide variety of structural properties and their corresponding measures have been proposed to quantify the disruption inflicted on the graph G. In general, these measures are either associated with (1) optimal solutions to flow problems over G, such as shortest paths, maximum flow, or minimum cost flow problems [17,19,33,38,41,44,49,78,79]; (2) the sizes or relative weight of some topological node or edge structures in G, like spanning trees [28], dominating sets [47], central nodes-for example, the one-median or one-center nodes [10], matchings [80,81], independent sets [9], cliques [46], and node covers [9]; and (3) connectivity and cohesiveness properties of G, such as the total number of connected node pairs [1, 6, 21-23, 51, 52, 65, 69, 70, 72, 73], the weight of the connections between the node pairs [6,21,73], the size of the largest component [13,30,55,63,64,73], the total number of components [5,[63][64][65]68], distance-based connectivity metrics [74], and the graph information entropy [14,35,56,73].…”

Section: Introductionmentioning

confidence: 99%

Detecting critical node structures on graphs: A mathematical programming approach

Walteros¹,

Veremyev

Pardalos

et al. 2018

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We consider the problem of detecting a collection of critical node structures of a graph whose deletion results in the maximum deterioration of the graph's connectivity. The proposed approach is aimed to generalize other existing models whose scope is restricted to removing individual and unrelated nodes. We consider two common metrics to quantify the connectivity of the residual graph: the total number of connected node pairs and the size of the largest connected component. We first discuss the computational complexity of the problem and then introduce a general mixed-integer linear formulation, which depending on the kind of node structures, may have an exponentially large number of variables and constraints. To solve this potentially large model, we develop a branch-price-and-cut framework, along with some valid inequalities and preprocessing algorithms to strengthen the formulation and reduce the overall execution time. We use the proposed approach to solve the problem for the cases, where the node structures form cliques or stars and provide further directions on how to extend the framework for detecting other kinds of critical structures as well. Finally, we test the quality of our approach by solving a collection of real-life and randomly generated instances with various configurations, analyze the benefits of our model, and propose further enhancements. KEYWORDS branch-price-and-cut, combinatorial optimization, critical node problem, graph partitioning, mixed-integer programming, network interdiction Problem 2. Critical star detection problem (CSP). INSTANCE: A simple non-empty graph G = (V, E) and two nonnegative integers r and b. QUESTION: Is there a set  of at most b stars so that a connectivity measure over G[V ⧵ V()] is no greater than r. Theorem 2. CSP for both the CNP and LC measures is strongly NP-complete.The proof of this theorem, which can also be found in the Appendix A, uses a reduction from DOMINATING SET to CSP. Complexity of detecting critical connected subgraphsWe now prove a more general complexity result, in which the elements of set  induce connected subgraphs of G. We show that detecting critical disjoint connected subgraphs is NP-hard as well.Problem 3. Critical connected subgraph detection problem (CCSP). INSTANCE:A simple non-empty graph G = (V, E), a set of connected subgraphs  , and two nonnegative integers r and b.

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Section: Introductionmentioning

confidence: 99%

Detecting critical node structures on graphs: A mathematical programming approach

Walteros¹,

Veremyev

Pardalos

et al. 2018

Networks

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“…See, for instance, Shen et al (2012), Granata et al (2013), Veremyev, Boginski and Pasiliao (2014), Veremyev, Prokopyev and Pasiliao (2014), a survey in Walteros and Pardalos (2012) and the references given therein.…”

Section: Introductionmentioning

confidence: 99%

Sequential Shortest Path Interdiction with Incomplete Information

2016

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We study sequential interdiction when the interdictor has incomplete initial information about the network, and the evader has complete knowledge of the network, including its structure and arc costs. In each time period, the interdictor blocks at most k arcs from the network observed up to that period, after which the evader travels along a shortest path between two (fixed) nodes in the interdicted network. By observing the evader's actions, the interdictor learns about the network structure and arc costs, and adjusts its actions so as to maximize the cumulative cost incurred by the evader. A salient feature of our work is that the feedback in each period is deterministic and adversarial. In addition to studying the regret minimization problem, we also discuss time-stability of a policy, which is the number of time periods until the interdictor's actions match those of an oracle interdictor with prior knowledge of the network. We propose a class of simple interdiction policies that have a finite regret and detect when the instantaneous regret reaches zero in real time. More importantly, we establish that this class of policies belongs to the set of efficient policies.

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“…Rocco and Ramirez‐Marquez present an evolutionary optimization approach to minimize the maximum flow in the network. Granata et al use branch and price algorithms to solve a version of the network interdiction problem.…”

Section: Problem Statementmentioning

confidence: 99%

Two extended formulations for cardinality maximum flow network interdiction problem

Rad

Kakhki

2017

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We consider the maximum flow network interdiction problem in its cardinality case. There is an integer programming model for this problem by Wood (Math Comput Model 17 (1993), 1–18). Two types of valid inequalities have also been proposed (Altner et al., Oper Res Lett 38 (2010), 33–38 and Wood, Math Comput Model 17 (1993), 1–18) to strengthen the LP relaxation of the integer model. However, due to their combinatorial nature, the number of these inequalities are exponential. Here, we present an equivalent reformulation (extended formulation) for this problem which has a polynomial number of constraints. We also introduce new valid inequalities, and show that the corresponding reformulation of the LP relaxation of the integer model augmented with these inequalities, significantly decreases the integrality gap for a class of network interdiction problems with proven large integrality gaps. Numerical results for some benchmark as well as randomly generated instances are also reported. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 367–377 2017

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Network interdiction via a Critical Disruption Path: Branch-and-Price algorithms

Cited by 22 publications

References 16 publications

Detecting critical node structures on graphs: A mathematical programming approach

Detecting critical node structures on graphs: A mathematical programming approach

Sequential Shortest Path Interdiction with Incomplete Information

Two extended formulations for cardinality maximum flow network interdiction problem

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