Optimization Letters

, Volume 8, Issue 4, pp 1245–1259 | Cite as

Exact identification of critical nodes in sparse networks via new compact formulations

  • Alexander Veremyev
  • Vladimir Boginski
  • Eduardo L. Pasiliao
Original Paper


Critical node detection problems aim to optimally delete a subset of nodes in order to optimize or restrict a certain metric of network fragmentation. In this paper, we consider two network disruption metrics which have recently received substantial attention in the literature: the size of the remaining connected components and the total number of node pairs connected by a path. Exact solution methods known to date are based on linear 0–1 formulations with at least \(\varTheta (n^3)\) entities and allow one to solve these problems to optimality only in small sparse networks with up to 150 nodes. In this work, we develop more compact linear 0–1 formulations for the considered types of problems with \(\varTheta (n^2)\) entities. We also provide reformulations and valid inequalities that improve the performance of the developed models. Computational experiments show that the proposed formulations allow finding exact solutions to the considered problems for real-world sparse networks up to 10 times larger and with CPU time up to 1,000 times faster compared to previous studies.


Combinatorial optimization Networks Graphs Critical nodes  Node interdiction Linear 0–1 formulations 



This work has been supported in part by the Air Force Research Laboratory (AFRL) and the Defense Threat Reduction Agency (DTRA). The research was performed while the first author held a National Research Council Research Associateship Award at AFRL. This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander Veremyev
    • 1
  • Vladimir Boginski
    • 2
  • Eduardo L. Pasiliao
    • 1
  1. 1.Munitions DirectorateAir Force Research LaboratoryEglin AFBUSA
  2. 2.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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