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International Conference on Mathematical Optimization Theory and Operations Research

MOTOR 2020: Mathematical Optimization Theory and Operations Research pp 69–84Cite as

Using Integer Programming to Search for Counterexamples: A Case Study

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12095))

Abstract

It is known that there exist 4-regular, 1-tough graphs which are non-hamiltonian. The smallest such graph known has \(n=18\) nodes and was found by Bauer et al., who conjectured that all 4-regular, 1-tough graphs with \(n\le 17\) are hamiltonian. They in fact proved that this is true for \(n\le 15\), but left open the possibility of non-hamiltonian graphs of 16 or 17 nodes. By using ILP for modeling a counterexample, and then finding out that the model has no solutions, we give an algorithmic proof that their conjecture was indeed correct.

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References

  1. Appel, K., Haken, W.: Every planar map is four colorable. I. Discharging. Illinois J. Math. 21(3), 429–490 (1977)

    Article  MathSciNet  Google Scholar 

  2. Bauer, D., Hakimi, S.L., Schmeichel, E.: Recognizing tough graphs is NP-hard. Discrete Appl. Math 28, 191–195 (1990)

    Article  MathSciNet  Google Scholar 

  3. Bauer, D., Broersma, H.J., Schmeichel, E.: Toughness in graphs - a survey. Graphs Comb. 22, 1–35 (2006)

    Article  MathSciNet  Google Scholar 

  4. Bauer, D., Broersma, H.J., Veldman, H.J.: On smallest nonhamiltonian regular tough graphs. Congressus Numerantium 70, 95–98 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Bauer, D., Broersma, H.J., Veldman, H.J.: Not every 2-tough graph is hamiltonian. Discrete Appl. Math 99, 317–321 (2000)

    Article  MathSciNet  Google Scholar 

  6. Broersma, H.J.: How tough is toughness? Bull. Eur. Assoc. Theoret. Comput. Sci. 117, 28–52 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Chvátal, V.: Tough graphs and hamiltonian circuits. Discrete Math. 5, 215–228 (1973)

    Article  MathSciNet  Google Scholar 

  8. DeLeon, M.: A study of sufficient conditions for hamiltonian cycles. Rose-Hulman Und. Math. J. 1(1), 1–19 (2000)

    Google Scholar 

  9. Desrosiers, J., Lübbecke, M.E.: Branch-Price-and-Cut Algorithms, in the Wiley Encyclopedia of Operations Research and Management Science. Wiley, Chichester (2010)

    Google Scholar 

  10. Gamrath, G., et al.: The SCIP Optimization Suite 3.2.1, ZIB-Report, pp. 15–60 (2016)

    Google Scholar 

  11. Hilbig, F.: Kantenstrukturen in nichthamiltonschen Graphen. Ph.D. thesis, Technische Universït at Berlin (1986)

    Google Scholar 

  12. Hoffman, A.J., Singleton, R.R.: On Moore graphs of diameter two and three. IBM J. Res. Dev. 4, 497–504 (1960)

    Article  Google Scholar 

  13. Margot, F.: Symmetry in integer linear programming. In: Jünger, M., et al. (eds.) 50 Years of Integer Programming 1958-2008, pp. 647–686. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-540-68279-0_17

    Chapter  MATH  Google Scholar 

  14. McKay, B.D., Piperno, A.: Practical graph isomorphism, II. J. Symbolic Comput. 60, 94–112 (2014)

    Article  MathSciNet  Google Scholar 

  15. McKay, B.D., Piperno, A.: nauty and Traces User’s Guide (Version 2.6) (2016)

    Google Scholar 

  16. Mitchell, J.E.: Branch-and-cut algorithms for combinatorial optimization problems. In: Pardalos, P.M., Resende, M.G.C. (eds.) Handbook of Applied Optimization, pp. 65–77. Oxford University Press, Oxford (2002)

    Google Scholar 

  17. Ostrowski, J., Linderoth, J., Rossi, F., Smriglio, S.: Orbital branching. Math. Program. 126(1), 147–178 (2011)

    Article  MathSciNet  Google Scholar 

  18. Pfetsch, M.E., Rehn, T.: A computational comparison of symmetry handling methods for mixed integer programs (2015). http://www.optimization-online.org

  19. Skiena, S.S.: The Algorithm Design Manual. Springer, London (2008). https://doi.org/10.1007/978-1-84800-070-4

    Book  MATH  Google Scholar 

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Funding

This research has been carried out in the framework of the departmental research project ICON: Innovative Combinatorial Optimization in Networks, Department of Mathematics, Computer Science and Physics (PRID 2017–2018), University of Udine, Italy.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche, Informatiche e Fisiche, University of Udine, Via delle Scienze 206, 33100, Udine, Italy

    Giuseppe Lancia, Eleonora Pippia & Franca Rinaldi

  2. The Research Hub by Electrolux Professional, Viale Treviso 15, 33170, Pordenone, PN, Italy

    Eleonora Pippia

Authors
  1. Giuseppe Lancia
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  2. Eleonora Pippia
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  3. Franca Rinaldi
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Corresponding author

Correspondence to Giuseppe Lancia .

Editor information

Editors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

    Alexander Kononov

  2. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  3. National Research University Higher School of Economics, Nizhny Novgorod, Russia

    Valery A Kalyagin

  4. University of Florida, Gainesville, FL, USA

    Panos Pardalos

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Lancia, G., Pippia, E., Rinaldi, F. (2020). Using Integer Programming to Search for Counterexamples: A Case Study. In: Kononov, A., Khachay, M., Kalyagin, V., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2020. Lecture Notes in Computer Science(), vol 12095. Springer, Cham. https://doi.org/10.1007/978-3-030-49988-4_5

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