Polynomial-time algorithms for solving a class of critical node problems on trees and series-parallel graphs

  • Authors:

  • Siqian Shen;J. Cole Smith

  • Affiliations:

  • Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, P.O. Box 116595, Gainesville, Florida 32611-6595;Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, P.O. Box 116595, Gainesville, Florida 32611-6595

  • Venue:
  • Networks
  • Year:
  • 2012

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Abstract

We examine variants of the critical node problem on specially structured graphs, which aim to identify a subset of nodes whose removal will maximally disconnect the graph. These problems lie at the intersection of network interdiction and graph theory research and are relevant to several practical optimization problems. The two different connectivity metrics that we consider regard the number of maximal connected components (which we attempt to maximize) and the largest component size (which we attempt to minimize). We develop optimal polynomial-time dynamic programming algorithms for solving these problems on tree structures and on series-parallel graphs, corresponding to each graph-connectivity metric. We also extend our discussion by considering node deletion costs, node weights, and solving the problems on generalizations of tree structures. Finally, we demonstrate the computational efficacy of our approach on randomly generated graph instances. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012 © 2012 Wiley Periodicals, Inc.