Branch and cut algorithms for detecting critical nodes in undirected graphs

In this paper we deal with the critical node problem, where a given number of nodes has to be removed from an undirected graph in order to maximize the disconnections between the node pairs of the graph. We propose an integer linear programming model with a non-polynomial number of constraints but whose linear relaxation can be solved in polynomial time. We derive different valid inequalities and some theoretical results about them. We also propose an alternative model based on a quadratic reformulation of the problem. Finally, we perform many computational experiments and analyze the corresponding results.

[1]  J. C. Smith,et al.  Algorithms for discrete and continuous multicommodity flow network interdiction problems , 2007 .

[2]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

[3]  Clayton W. Commander,et al.  Identifying Critical Nodes in Protein-Protein Interaction Networks , 2009 .

[4]  Hanif D. Sherali,et al.  A reformulation-convexification approach for solving nonconvex quadratic programming problems , 1995, J. Glob. Optim..

[5]  B. Borchers.  CSDP, A C library for semidefinite programming , 1999 .

[6]  P. Pardalos,et al.  Clustering challenges in biological networks , 2009 .

[7]  B. Borchers.  A C library for semidefinite programming , 1999 .

[8]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[9]  László Lovász,et al.  Graph Theory and Integer Programming , 1979 .

[10]  R. W. Lucky,et al.  Free software [Reflections] , 1999 .

[11]  Panos M. Pardalos,et al.  Robust Optimization of Graph Partitioning and Critical Node Detection in Analyzing Networks , 2010, COCOA.

[12]  R. Kevin Wood,et al.  Deterministic network interdiction , 1993 .

[13]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[14]  Panos M. Pardalos,et al.  Detecting critical nodes in sparse graphs , 2009, Comput. Oper. Res..

[15]  Taieb Znati,et al.  On Approximation of New Optimization Methods for Assessing Network Vulnerability , 2010, 2010 Proceedings IEEE INFOCOM.

[16]  Alan T. Murray,et al.  Modeling s-t path availability to support disaster vulnerability assessment of network infrastructure , 2010, Comput. Oper. Res..

[17]  Kurt M. Anstreicher,et al.  Institute for Mathematical Physics Semidefinite Programming versus the Reformulation–linearization Technique for Nonconvex Quadratically Constrained Quadratic Programming Semidefinite Programming versus the Reformulation-linearization Technique for Nonconvex Quadratically Constrained , 2022 .

[18]  Richard D. Wollmer,et al.  Removing Arcs from a Network , 1964 .

[19]  Hanif D. Sherali,et al.  Enhancing RLT-based relaxations for polynomial programming problems via a new class of v-semidefinite cuts , 2012, Comput. Optim. Appl..

[20]  Hanif D. Sherali,et al.  Enhancing RLT relaxations via a new class of semidefinite cuts , 2002, J. Glob. Optim..

[21]  Hanif D. Sherali,et al.  A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique , 1992, J. Glob. Optim..

[22]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-integer Zero-one Programming Problems , 1994, Discret. Appl. Math..