# Branch and cut algorithms for detecting critical nodes in undirected graphs

- 523 Downloads
- 28 Citations

## Abstract

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.

## Keywords

Critical node problem Branch and cut Valid inequalities Reformulation-linearization technique## Notes

### Acknowledgements

The authors would like to thank two anonymous referees, whose constructive comments helped to improve the initial version of the paper.

## References

- 1.Anstreicher, K.M.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim.
**43**, 471–484 (2009) MathSciNetzbMATHCrossRefGoogle Scholar - 2.Arulselvan, A., Commander, C.W., Elefteriadou, L., Pardalos, P.M.: Detecting critical nodes in sparse graphs. Comput. Oper. Res.
**36**, 2193–2200 (2009) MathSciNetzbMATHCrossRefGoogle Scholar - 3.Boginski, V., Commander, C.W.: Identifying critical nodes in protein–protein interaction networks. In: Butenko, S., Art Chaovalitwongse, W., Pardalos, P.M. (eds.) Clustering Challenges in Biological Networks, pp. 153–167. World Scientific, Singapore (2009) CrossRefGoogle Scholar
- 4.Borchers, B.: CSDP, a C library for semidefinite programming. Optim. Methods Softw.
**11**, 613–623 (1999) MathSciNetCrossRefGoogle Scholar - 5.Christof, T.: PORTA—a Polyhedron Representation Transformation Algorithm. Free software. Revised by A. Löbel Google Scholar
- 6.Dinh, T.N., Xuan, Y., Thai, M.T., Park, E.K., Znati, T.: On approximation of new optimization methods for assessing network vulnerability. In: Proceedings of the 29th IEEE Conference on Computer Communications (INFOCOM), pp. 105–118 (2010) Google Scholar
- 7.Dreier, D.: Barabasi graph generator v1.4. http://www.cs.ucr.edu/ddreier
- 8.Fan, N., Pardalos, P.M.: Robust optimization of graph partitioning and critical node detection in analyzing networks. In: COCOA, 2010, pp. 170–183 (2010) Google Scholar
- 9.Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer, Berlin (2000) zbMATHGoogle Scholar
- 10.Lovász, L.: Graph theory and integer programming. Ann. Discrete Math.
**4**, 141–158 (1979) MathSciNetzbMATHCrossRefGoogle Scholar - 11.Matisziw, T.C., Murray, A.T.: Modeling s-t path availability to support disaster vulnerability assessment of network infrastructure. Comput. Oper. Res.
**36**, 16–26 (2009) zbMATHCrossRefGoogle Scholar - 12.Myung, Y.-S., Kim, H.-J.: A cutting plane algorithm for computing k-edge survivability of a network. Eur. J. Oper. Res.
**156**(3), 579–589 (2004) MathSciNetzbMATHCrossRefGoogle Scholar - 13.Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math.
**3**, 411–430 (1990) MathSciNetzbMATHCrossRefGoogle Scholar - 14.Sherali, H.D., Fraticelli, B.M.: Enhancing RLT relaxations via a new class of semidefinite cuts. J. Glob. Optim.
**22**, 233–261 (2002) MathSciNetzbMATHCrossRefGoogle Scholar - 15.Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Glob. Optim.
**2**, 101–112 (1992) MathSciNetzbMATHCrossRefGoogle Scholar - 16.Sherali, H.D., Tuncbilek, C.H.: A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Glob. Optim.
**7**, 1–31 (1995) MathSciNetzbMATHCrossRefGoogle Scholar - 17.Sherali, H.D., Dalkiran, E., Desai, J.: Enhancing RLT-based relaxations for polynomial programming problems via a new class of
*ν*-semidefinite cuts. Comput. Optim. Appl. (2011). doi: 10.1007/s10589-011-9425-z Google Scholar - 18.Smith, J.C., Lim, C.: Algorithms for discrete and continuous multicommodity flow network interdiction problems. IIE Trans.
**39**, 15–26 (2007) CrossRefGoogle Scholar - 19.Wollmer, R.: Removing arcs from a network. Oper. Res.
**12**, 934–940 (1964) MathSciNetzbMATHCrossRefGoogle Scholar - 20.Wood, R.K.: Deterministic network interdiction. Math. Comput. Model.
**17**, 1–18 (1993) zbMATHCrossRefGoogle Scholar