Skip to main content

On the Complexity of the Minimum Chromatic Violation Problem

  • Conference paper
  • First Online:
Combinatorial Optimization (ISCO 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14594))

Included in the following conference series:

  • 339 Accesses

Abstract

In this paper, we consider a generalization of the classical vertex coloring problem of a graph, where the edge set of the graph is partitioned into strong and weak edges; the endpoints of a weak edge can be assigned to the same color and the minimum chromatic violation problem (MCVP) asks for a coloring of the graph minimizing the number of weak edges having its endpoints assigned to the same color. Previous works in the literature on MCVP focus on defining integer programming formulations and performing polyhedral studies on the associated polytopes but, to the best of our knowledge, very few computational complexity studies exist for MCVP. In this work, we focus on the computational complexity of this problem over several graph families such as interval and unit interval graphs, among others. We show that MCVP is NP-hard for general graphs and it remains NP-hard when the graph induced by the strong edges is unit interval or distance hereditary. On the other side, we provide a polynomial algorithm that properly solves MCVP when the graph is a unit interval graph without triangles with two or more weak edges.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Hoboken (1993)

    Google Scholar 

  2. Biró, M., Hujter, M., Tuza, Z.S.: Precoloring extension. I. Interval graphs. Discrete Math. 100, 267–279 (1992)

    Article  MathSciNet  Google Scholar 

  3. Borndörfer, R., Eisenblätter, A., Grötschel, M., Martin, A.: The Orientation Model for Frequency Assignment Problems, ZIB-Berlin TR 98–01 (1998)

    Google Scholar 

  4. Bonomo, F., Durán, G., Marenco, J.: Exploring the complexity boundary between coloring and list-coloring. Ann. Oper. Res. 169, 3–16 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bonomo-Braberman, F., Gonzalez, C.: A new approach on locally checkable problems. Discret. Appl. Math. 314, 53–80 (2022)

    Article  MathSciNet  Google Scholar 

  6. Braga, M., Delle Donne, D., Escalante, M.S., Marenco, J., Ugarte, M.E., Varaldo, M.C.: The minimum chromatic violation problem: a polyhedral approach. Discret. Appl. Math. 281, 69–80 (2020)

    Article  MathSciNet  Google Scholar 

  7. Burke, E., Marecek, J., Parkes, A., Rudová, H.: A supernodal formulation of vertex colouring with applications in course timetabling. Ann. Oper. Res. 179–1, 105–130 (2010)

    Article  MathSciNet  Google Scholar 

  8. Carlisle, M.C., Lloyd, E.L.: On the k-coloring of intervals. Discret. Appl. Math. 59, 225–235 (1995)

    Article  Google Scholar 

  9. Chow, F.C., Hennessy, J.L.: The priority-based coloring approach to register allocation. ACM Trans. Program. Lang. Syst. 12–4, 501–536 (1990)

    Article  Google Scholar 

  10. Delle Donne, D., Escalante, M.S., Ugarte, M.E.: Implementing cutting planes for the chromatic violation problem. In: Proceedings of the Joint ALIO/EURO International Conference 2021–2022 on Applied Combinatorial Optimization, OpenProceedings.org, pp. 17–22 (2022)

    Google Scholar 

  11. de Werra, D.: An introduction to timetabling. Eur. J. Oper. Res. 19, 151–162 (1985)

    Article  MathSciNet  Google Scholar 

  12. Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. Assoc. Comput. Mach. 19, 248–264 (1972)

    Article  Google Scholar 

  13. Gamst, A.: Some lower bounds for a class of frequency assignment problems. IEEE Trans. Veh. Technol. 35–1, 8–14 (1986)

    Article  Google Scholar 

  14. Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness (1979). Freeman, W. H

    Google Scholar 

  15. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)

    Article  MathSciNet  Google Scholar 

  16. Leighton, F.T.: A graph coloring algorithm for large scheduling problems. J. Res. Natl. Bur. Stand. 84–6, 489–503 (1979)

    Article  MathSciNet  Google Scholar 

  17. Marx, D.: Precoloring extension on unit interval graphs. Discret. Appl. Math. 154, 995–1002 (2006)

    Article  MathSciNet  Google Scholar 

  18. Woo, T.K., Su, S.Y.W., Newman Wolfe, R.: Resource allocation in a dynamically partitionable bus network using a graph coloring algorithm. IEEE Trans. Commun. 39–12, 1794–1801 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to María Elisa Ugarte .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Delle Donne, D., Escalante, M., Ugarte, M.E. (2024). On the Complexity of the Minimum Chromatic Violation Problem. In: Basu, A., Mahjoub, A.R., Salazar González, J.J. (eds) Combinatorial Optimization. ISCO 2024. Lecture Notes in Computer Science, vol 14594. Springer, Cham. https://doi.org/10.1007/978-3-031-60924-4_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-60924-4_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-60923-7

  • Online ISBN: 978-3-031-60924-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics