Soft Computing

, Volume 23, Issue 23, pp 12729–12744 | Cite as

The bi-objective critical node detection problem with minimum pairwise connectivity and cost: theory and algorithms

  • Juan Li
  • Panos M. Pardalos
  • Bin XinEmail author
  • Jie Chen
Methodologies and Application


An effective way to analyze and apprehend structural properties of networks is to find their most critical nodes. This paper studies a bi-objective critical node detection problem, denoted as Bi-CNDP. In this variant, we do not make any assumptions on the psychology of decision makers and seek to find a set of solutions which minimize the pairwise connectivity of the induced graph and the cost of removing these critical nodes at the same time. After explicitly stating the formulation of Bi-CNDP, we first prove the NP-hardness of this problem for general graphs and the existence of a polynomial algorithm for constructing the \(\varepsilon \)-approximated Pareto front for Bi-CNDPs on trees. Then different approaches of determining the mating pool and the replacement pool are proposed for the decomposition-based multi-objective evolutionary algorithms. Based on this, two types of decomposition-based multi-objective evolutionary algorithms (MOEA/D and DMOEA-\(\varepsilon \hbox {C}\)) are modified and applied to solve the proposed Bi-CNDP. Numerical experiments on sixteen famous benchmark problems with random and logarithmic weights are firstly conducted to assess different types of the mating pool and the replacement pool. Besides, computational results between two improved algorithms, i.e., I-MOEA/D and I-DMOEA-\(\varepsilon \hbox {C}\), demonstrate that they behave differently on these instances and I-DMOEA-\(\varepsilon \hbox {C}\) shows better performance on the majority of test instances. Finally, a decision-making process from the perspective of minimizing the pairwise connectivity of the induced graph given a constraint on the cost of removing nodes is presented for helping decision makers to identify the most critical nodes for further protection or attack.


Bi-objective critical node detection problems NP-hardness \(\varepsilon \)-Approximation Decomposition-based multi-objective evolutionary algorithms Mating pool Replacement pool 



This study was funded by the National Outstanding Youth Talents Support Program 61822304, the NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization under Grant U1609214, the National Natural Science Foundation of China under Grant 61673058, the Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant 61621063, and the Projects of Major International (Regional) Joint Research Program NSFC under Grant 61720106011. P. M. Pardalos is supported by the Paul and Heidi Brown Preeminent Professorship in Industrial and Systems Engineering, University of Florida.

Compliance with ethical standards

Conflict of Interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Juan Li
    • 1
    • 2
    • 3
  • Panos M. Pardalos
    • 4
  • Bin Xin
    • 1
    • 2
    • 3
    Email author
  • Jie Chen
    • 1
    • 2
    • 3
  1. 1.School of Automation, Beijing Institute of TechnologyBeijingChina
  2. 2.State Key Laboratory of Intelligent Control and Decision of Complex SystemsBeijing Institute of TechnologyBeijingChina
  3. 3.Beijing Advanced Innovation Center for Intelligent Robots and SystemsBeijing Institute of TechnologyBeijingChina
  4. 4.Center for Applied Optimization, Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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