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Maximum Cut on Interval Graphs of Interval Count Four is NP-Complete

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Abstract

The computational complexity of the MaxCut problem restricted to interval graphs has been open since the 80’s, being one of the problems proposed by Johnson in his Ongoing Guide to NP-completeness, and has been settled as NP-complete only recently by Adhikary, Bose, Mukherjee, and Roy. On the other hand, many flawed proofs of polynomiality for MaxCut on the more restrictive class of unit/proper interval graphs (or graphs with interval count 1) have been presented along the years, and the classification of the problem is still unknown. In this paper, we present the first NP-completeness proof for MaxCut when restricted to interval graphs with bounded interval count, namely graphs with interval count 4.

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References

  1. Adhikary, R., Bose, K., Mukherjee, S., Roy, B.: Complexity of maximum cut on interval graphs. In: 37th International Symposium on Computational Geometry. Leibniz Int. Proc. Inform., vol. 189, # 7. Leibniz-Zent. Inform., Wadern (2021)

  2. Berman, P., Karpinski, M.: On some tighter inapproximability results. In: Automata, Languages and Programming (Prague 1999). Lecture Notes in Computer Science, vol. 1644, pp. 200–209. Springer, Berlin (1999)

  3. Bodlaender, H.L., de Figueiredo, C.M.H., Gutierrez, M., Kloks, T., Niedermeier, R.: Simple max-cut for split-indifference graphs and graphs with few \(P_4\)’s. In: 3rd International Workshop on Experimental and Efficient Algorithms (Angra dos Reis 2004). Lecture Notes in Computer Science, vol. 3059, pp. 87–99. Springer, Berlin (2004)

  4. Bodlaender, H.L., Kloks, T., Niedermeier, R.: Simple max-cut for unit interval graphs and graphs with few \(P_4\)s. In: 6th Twente Workshop on Graphs and Combinatorial Optimization (Enschede 1999). Electronic Notes in Discrete Mathematics, vol. 3, pp. 19–26. Elsevier, Amsterdam (1999)

  5. Bondy, J.A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics, vol. 244. Springer, New York (2008)

  6. Boyacı, A., Ekim, T., Shalom, M.: A polynomial-time algorithm for the maximum cardinality cut problem in proper interval graphs. Inform. Process. Lett. 121, 29–33 (2017)

    Article  MathSciNet  Google Scholar 

  7. Cerioli, M.R., de Oliveira, F.S., Szwarcfiter, J.L.: On counting interval lengths of interval graphs. Discrete Appl. Math. 159(7), 532–543 (2011)

    Article  MathSciNet  Google Scholar 

  8. Cerioli, M.R., de Oliveira, F.S., Szwarcfiter, J.L.: The interval count of interval graphs and orders: a short survey. J. Braz. Comput. Soc. 18(2), 103–112 (2012)

    Article  MathSciNet  Google Scholar 

  9. Chakraborty, D., Das, S., Foucaud, F., Gahlawat, H., Lajou, D., Roy, B.: Algorithms and complexity for geodetic sets on planar and chordal graphs. In: 31st International Symposium on Algorithms and Computation. Leibniz Int. Proc. Inform., vol. 181, # 7. Leibniz-Zent. Inform., Wadern (2020)

  10. Cohen, J., Fomin, F., Heggernes, P., Kratsch, D., Kucherov, G.: Optimal linear arrangement of interval graphs. In: International Symposium on Mathematical Foundations of Computer Science (Stará Lesná 2006). Lecture Notes in Computer Science, vol. 4162, pp. 267–279. Springer, Berlin (2006)

  11. Corneil, D.G., Kim, H., Natarajan, S., Olariu, S., Sprague, A.P.: Simple linear time recognition of unit interval graphs. Inform. Process. Lett. 55(2), 99–104 (1995)

    Article  MathSciNet  Google Scholar 

  12. Ekim, T., Erey, A., Heggernes, P., van ’t Hof, P., Meister, D.: Computing minimum geodetic sets of proper interval graphs. LATIN 2012: Theoretical Informatics (Arequipa 2012). Lecture Notes in Computer Science, vol. 7256, pp. 279–290. Springer, Heidelberg (2012)

  13. de Figueiredo, C.M.H., de Melo, A.A., Oliveira, F.S., Silva, A.: Maximum cut on interval graphs of interval count five is NP-complete (2020). arXiv:2012.09804

  14. de Figueiredo, C.M.H., de Melo, A.A., Oliveira, F.S., Silva, A.: Maximum cut on interval graphs of interval count four is NP-complete. In: 46th International Symposium on Mathematical Foundations of Computer Science (Tallinn 2021). Leibniz Int. Proc. Inform., vol. 202, # 38. Leibniz-Zent. Inform., Wadern (2021)

  15. de Figueiredo, C.M.H., de Melo, A.A., Sasaki, D., Silva, A.: Revising Johnson’s table for the 21st century. Discrete Appl. Math. 323, 184–200 (2022)

  16. Fishburn, P.C.: Interval graphs and interval orders. Discrete Math. 55(2), 135–149 (1985)

    Article  MathSciNet  Google Scholar 

  17. Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified \({N\!P}\)-complete problems. Theor. Comput. Sci. 1(3), 237–267 (1976)

    Article  Google Scholar 

  18. Herrera de Figueiredo, C.M., Meidanis, J., Picinin de Mello, C.: A linear-time algorithm for proper interval graph recognition. Inform. Process. Lett. 56(3), 179–184 (1995)

  19. Johnson, D.S.: The NP-completeness column: an ongoing guide. J. Algorithms 6(3), 434–451 (1985)

    Article  MathSciNet  Google Scholar 

  20. Klavík, P., Otachi, Y., Šejnoha, J.: On the classes of interval graphs of limited nesting and count of lengths. Algorithmica 81(4), 1490–1511 (2019)

    Article  MathSciNet  Google Scholar 

  21. Kratochvíl, J., Masařík, T., Novotná, J.: \(\cal{U}\)-bubble model for mixed unit interval graphs and its applications: the MaxCut problem revisited. In: 45th International Symposium on Mathematical Foundations of Computer Science. Leibniz Int. Proc. Inform., vol. 170, # 57. Leibniz-Zent. Inform., Wadern (2020)

  22. Marx, D.: A short proof of the NP-completeness of minimum sum interval coloring. Oper. Res. Lett. 33(4), 382–384 (2005)

    Article  MathSciNet  Google Scholar 

  23. Nicoloso, S., Sarrafzadeh, M., Song, X.: On the sum coloring problem on interval graphs. Algorithmica 23(2), 109–126 (1999)

    Article  MathSciNet  Google Scholar 

  24. Roberts, F.S.: Indifference graphs. In: Proof Techniques in Graph Theory (Ann Arbor 1968), pp. 139–146. Academic Press, New York (1969)

  25. Yuan, J., Zhou, S.: Optimal labelling of unit interval graphs. Appl. Math. J. Chin. Univ. Ser. B 10(3), 337–344 (1995)

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Acknowledgements

We thank Vinicius F. Santos who shared reference [1], and anonymous referees for many valuable suggestions, including improving the interval count from 5 to 4.

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Authors and Affiliations

  1. Programa de Engenharia de Sistemas e Computção, Universidade Federal do Rio de Janeiro, Caixa Postal 68511, Rio de Janeiro, Rio de Janeiro, 21941-972, Brazil

    Celina M. H. de Figueiredo & Alexsander A. de Melo

  2. Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier, 524, Rio de Janeiro, Rio de Janeiro, 20550-900, Brazil

    Fabiano S. Oliveira

  3. Departamento de Matemática - Centro de Ciências, Universidade Federal do Ceará, Av. Mister Hull s/n - Bloco 914, Fortaleza, Ceará, 60455-760, Brazil

    Ana Silva

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  1. Celina M. H. de Figueiredo
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  3. Fabiano S. Oliveira
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Correspondence to Ana Silva.

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de Figueiredo, C.M.H., de Melo, A.A., Oliveira, F.S. et al. Maximum Cut on Interval Graphs of Interval Count Four is NP-Complete. Discrete Comput Geom 71, 893–917 (2024). https://doi.org/10.1007/s00454-023-00508-x

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  • DOI: https://doi.org/10.1007/s00454-023-00508-x

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