Detecting critical nodes in sparse graphs

Authors:
Ashwin Arulselvan;Clayton W. Commander;Lily Elefteriadou;Panos M. Pardalos
Affiliations:
Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA;Air Force Research Laboratory, Munitions Directorate, and Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA;Department of Civil and Coastal Engineering, University of Florida, Gainesville, FL, USA;Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA
Venue:
Computers and Operations Research
Year:
2009

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Abstract

Identifying critical nodes in a graph is important to understand the structural characteristics and the connectivity properties of the network. In this paper, we focus on detecting critical nodes, or nodes whose deletion results in the minimum pair-wise connectivity among the remaining nodes. This problem, known as the critical node problem has applications in several fields including biomedicine, telecommunications, and military strategic planning. We show that the recognition version of the problem is NP-complete and derive a mathematical formulation based on integer linear programming. In addition, we propose a heuristic for the problem which exploits the combinatorial structure of the graph. The heuristic is then enhanced by the application of a local improvement method. A computational study is presented in which we apply the integer programming formulation and the heuristic to real and randomly generated data sets. For all instances tested, the heuristic is able to efficiently provide optimal solutions in a fraction of the time required by a commercial software package.