Iterated backtrack removal search for finding k-vertex-critical subgraphs


Given an undirected graph \(G = (V,E)\) and a positive integer k, a k-vertex-critical subgraph (k-VCS) of G is a subgraph H such that its chromatic number equals k (i.e., \(\chi (H) = k\)), and removing any vertex causes a decrease of \(\chi (H)\). The k-VCS problem (k-VCSP) is to find the smallest k-vertex-critical subgraph \(H^*\) of G. This paper proposes an iterated backtrack-based removal (IBR) heuristic to find k-VCS for a given graph G. IBR extends the popular removal strategy that is intensification-oriented. The proposed extensions include two new diversification-oriented search components—a backtracking mechanism to reconsider some removed vertices and a perturbation strategy to escape local optima traps. Computational results on 80 benchmark graphs show that IBR is very competitive in terms of solution quality and run-time efficiency compared with state-of-the-art algorithms in the literature. Specifically, IBR improves the best-known solutions for 9 graphs and matches the best results for other 70 instances. We investigate the interest of the key components of the proposed algorithm.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

  2. 2.

  3. 3.



  1. Chakravarti, N.: Some results concerning post-infeasibility analysis. Eur. J. Oper. Res. 73(1), 139–143 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  2. Chinneck, J.W.: Minos (IIS): infeasibility analysis using minos. Comput. Oper. Res. 21(1), 1–9 (1994)

    Article  MATH  Google Scholar 

  3. Chinneck, J.W.: Finding a useful subset of constraints for analysis in an infeasible linear program. INFORMS J. Comput. 9(2), 164–174 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  4. Chinneck, J.W., Dravnieks, E.W.: Locating minimal infeasible constraint sets in linear programs. ORSA J. Comput. 3(2), 157–168 (1991)

    Article  MATH  Google Scholar 

  5. Desrosiers, C., Galinier, P., Hertz, A.: Efficient algorithms for finding critical subgraphs. Discrete Appl. Math. 156(2), 244–266 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  6. Desrosiers, C., Galinier, P., Hertz, A., Paroz, S.: Using heuristics to find minimal unsatisfiable subformulas in satisfiability problems. J. Comb. Optim. 18(2), 124–150 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  7. Dorne, R., Hao, J.-K.: A new genetic local search algorithm for graph coloring. Lect. Notes Comput. Sci. 1498, 745–754 (1998)

    Article  Google Scholar 

  8. Eisenberg, C., Faltings, B.: Using the breakout algorithm to identify hard and unsolvable subproblems. Lect. Notes Comput. Sci. 2833, 822–826 (2003)

    Article  Google Scholar 

  9. Galinier, P., Hao, J.-K.: Hybrid evolutionary algorithms for graph coloring. J. Comb. Optim. 3(4), 379–397 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  10. Galinier, P., Hertz, A., Zufferey, N.: An adaptive memory algorithm for the k-colouring problem. Discrete Appl. Math. 156(2), 267–279 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  11. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness, vol. 58. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  12. Glover, F.: Tabu search—part I. ORSA J. Comput. 1(3), 190–206 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  13. Glover, F.: Tabu search—part II. ORSA J. Comput. 2(1), 4–32 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  14. Glover, F., Lü, Z., Hao, J.-K.: Diversification-driven tabu search for unconstrained binary quadratic problems. 4OR Q. J. Oper. Res. 8(3), 239–253 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  15. Grégoire, E., Mazure, B., Piette, C.: On finding minimally unsatisfiable cores of CSPs. Int. J. Artif. Intell. Tools 17(4), 745–763 (2008)

    Article  Google Scholar 

  16. Herrmann, F., Hertz, A.: Finding the chromatic number by means of critical graphs. J. Exp. Algorithmics JEA 7, 10 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  17. Hertz, A., de Werra, D.: Using tabu search techniques for graph coloring. Computing 39(4), 345–351 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  18. Johnson, D.S., Aragon, C.R., McGeoch, L.A., Schevon, C.: Optimization by simulated annealing: an experimental evaluation; part I, graph partitioning. Oper. Res. 37(6), 865–892 (1989)

    Article  MATH  Google Scholar 

  19. Liffiton, M.H., Sakallah, K.A.: On finding all minimally unsatisfiable subformulas. In: International Conference on Theory and Applications of Satisfiability Testing, pp. 173–186. Springer (2005)

  20. Liffiton, M.H., Sakallah, K.A.: Algorithms for computing minimal unsatisfiable subsets of constraints. J. Autom. Reason. 40(11), 133 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Lü, Z., Hao, J.-K.: A critical element-guided perturbation strategy for iterated local search. In: Evolutionary Computation in Combinatorial Optimization, pp. 1–12. Springer (2009)

  22. Lynce, I., Marques-Silva, J.P.: On computing minimum unsatisfiable cores. In: Online Proceedings of The Seventh International Conference on Theory and Applications of Satisfiability Testing (SAT 2004), 10–13 May 2004, Vancouver, BC, Canada (2004)

  23. Mneimneh, M., Lynce, I., Andraus, Z., Marques-Silva, J., Sakallah, K.: A branch-and-bound algorithm for extracting smallest minimal unsatisfiable formulas. In: International Conference on Theory and Applications of Satisfiability Testing, pp. 467–474. Springer (2005)

  24. Oh, Y., Mneimneh, M.N., Andraus, Z.S., Sakallah, K.A., Markov, I.L.: Amuse: a minimally-unsatisfiable subformula extractor. In: Proceedings of the 41st Annual Design Automation Conference, pp. 518–523. ACM (2004)

  25. Tamiz, M., Mardle, S.J., Jones, D.F.: Detecting IIS in infeasible linear programmes using techniques from goal programming. Comput. Oper. Res. 23(2), 113–119 (1996)

    Article  MATH  Google Scholar 

  26. van Loon, J.N.M.: Irreducibly inconsistent systems of linear inequalities. Eur. J. Oper. Res. 8(3), 283–288 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  27. Zhou, Z., Li, C.M., Huang, C., Xu, R.: An exact algorithm with learning for the graph coloring problem. Comput. Oper. Res. 51, 282–301 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  28. Zhou, Y., Hao, J.-K., Glover, F.: Memetic search for identifying critical nodes in sparse graphs (2017a). arXiv:1705.04119

  29. Zhou, Y., Hao, J.-K., Duval, B.: When data mining meets optimization: a case study on the quadratic assignment problem (2017b). arXiv:1708.05214

Download references


We are grateful to Dr. Chumin Li for providing us with the code of Zhou et al. (2014). Support for the first author of this work from the China Scholarship Council (2015–2019) is also acknowledged.

Author information



Corresponding author

Correspondence to Jin-Kao Hao.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sun, W., Hao, JK. & Caminada, A. Iterated backtrack removal search for finding k-vertex-critical subgraphs. J Heuristics 25, 565–590 (2019).

Download citation


  • Vertex-critical subgraph
  • Graph coloring
  • Tabu search
  • Backtracking-based diversification
  • Irreducibly inconsistent systems