Skip to main content
SpringerLink
Log in

Fractional Path Coloring in Bounded Degree Trees with Applications

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

This paper studies the natural linear programming relaxation of the path coloring problem. We prove constructively that finding an optimal fractional path coloring is Fixed Parameter Tractable (FPT), with the degree of the tree as parameter: the fractional coloring of paths in a bounded degree trees can be done in a time which is linear in the size of the tree, quadratic in the load of the set of paths, while exponential in the degree of the tree. We give an algorithm based on the generation of an efficient polynomial size linear program. Our algorithm is able to explore in polynomial time the exponential number of different fractional colorings, thanks to the notion of trace of a coloring that we introduce. We further give an upper bound on the cost of such a coloring in binary trees and extend this algorithm to bounded degree graphs with bounded treewidth. Finally, we also show some relationships between the integral and fractional problems, and derive a 1+5/3e≈1.61—approximation algorithm for the path coloring problem in bounded degree trees, improving on existing results. This classic combinatorial problem finds applications in the minimization of the number of wavelengths in wavelength division multiplexing (wdm) optical networks.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auletta, V., Caragiannis, I., Kaklamanis, C., Persiano, P.: Randomized path coloring on binary trees. In: APPROX’00. Lecture Notes in Computer Science, vol. 1913, pp. 60–71. Springer, Berlin (2000)

    Google Scholar 

  2. Azuma, K.: Weighted sum of certain dependent random variables. Tohoku Math. J. 19, 357–367 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bermond, J.-C., Gargano, L., Pérennes, S., Rescigno, A.A., Vaccaro, U.: Efficient collective communication in optical networks. In: ICALP’96. Lecture Notes in Computer Science, vol. 1099, pp. 574–585. Springer, Berlin (1996)

    Google Scholar 

  4. Bondy, J.A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics Series, vol. 244. Springer, Berlin (2008)

    MATH  Google Scholar 

  5. Caragiannis, I., Kaklamanis, C.: Approximate path coloring with applications to wavelength assignment in WDM optical networks. In: Proceedings of the 21st International Symposium on Theoretical Aspects of Computer Science (STACS ’04). Lecture Notes in Computer Science, vol. 2996, pp. 258–269. Springer, Berlin (2004)

    Google Scholar 

  6. Caragiannis, I., Kaklamanis, C., Persiano, P., Sidiropoulos, A.: Fractional and integral coloring of locally-symmetric sets of paths on binary trees. In: Proceedings of the 1st Workshop on Approximation and On-line Algorithms (WAOA ’03). Lecture Notes in Computer Science, vol. 2909, pp. 81–94. Springer, Berlin (2003)

    Google Scholar 

  7. Caragiannis, I., Kaklamanis, C., Persiano, P.: Approximation algorithms for path coloring in trees. In: Efficient Approximation and Online Algorithms. Lecture Notes in Computer Science, vol. 3484, pp. 74–96. Springer, Berlin (2006)

    Chapter  Google Scholar 

  8. Chlamtac, I., Ganz, A., Karmi, G.: Lightpath communications: An approach to high bandwidth optical WAN’s. IEEE Trans. Commun. 40(7), 1171–1182 (1992)

    Article  Google Scholar 

  9. Erlebach, T., Jansen, K., Kaklamanis, C., Mihail, M., Persiano, P.: Optimal wavelength routing on directed fiber trees. Theor. Comput. Sci. 221(1–2), 119–137 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Feige, U., Kilian, J.: Zero knowledge and the chromatic number. J. Comput. Syst. Sci. 57(2), 187–199 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ferreira, A., Pérennes, S., Richa, A., Rivano, H., Stier, N.: On the design of multifiber WDM networks. In: AlgoTel’02, Mèze, France, May 2002, pp. 25–32

  12. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)

    Google Scholar 

  13. Garg, N.: Multicommodity flows and approximation algorithms. Ph.D. thesis, Indian Institute of Technology, Delhi, April (1994)

  14. Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18(1), 3–20 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gargano, L., Hell, P., Pérennes, S.: Colouring paths in directed symmetric trees with applications to WDM routing. In: ICALP’97. Lecture Notes in Computer Science, vol. 1256, pp. 505–515. Springer, Berlin (1997)

    Google Scholar 

  16. Golumbic, M.C., Jamison, R.E.: The edge intersection graphs of paths in a tree. J. Comb. Theory 38, 8–22 (1985)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2, 2nd corrected edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  19. Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problems. PWS-Kent, Boston (1997)

    Google Scholar 

  20. Jansen, K.: Approximate strong separation with application in fractional graph coloring and preemptive scheduling. In: Proceedings of the 19th International Symposium on Theoretical Aspects of Computer Science (STACS ’02). Lecture Notes in Computer Science, vol. 2285, pp. 100–111. Springer, Berlin (2002)

    Google Scholar 

  21. Karapetyan, I.A.: On coloring of arc graphs. Dokl. Akad. Nauk Armianskoi CCP 70(5), 306–311 (1980). In Russian

    MATH  MathSciNet  Google Scholar 

  22. König, D.: Graphok és matrixok. Mat. Fiz. Lapok 116–119 (1931)

  23. Kumar, V.: Approximating arc circular colouring and bandwidth allocation in all-optical ring networks. In: APPROX’98 (1998)

  24. Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  25. Niessen, T., Kind, J.: The round-up property of the fractional chromatic number for proper circular arc graphs, J. Graph Theory (1998)

  26. Tarjan, R.E.: Decomposition by clique separators. Discrete Math. 55(2), 221–232 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  27. Tucker, A.: Coloring a family of circular arcs. SIAM J. Appl. Math. 29(3), 493–502 (1975)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Technology Institute, 61 Riga Feraiou str., Patras, Greece

    I. Caragiannis & C. Kaklamanis

  2. MASCOTTE Project, CNRS-I3S-INRIA, B.P. 93, 06902, Sophia Antipolis, France

    A. Ferreira, S. Pérennes & H. Rivano

Authors
  1. I. Caragiannis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Kaklamanis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Pérennes
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. H. Rivano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Rivano.

Additional information

A preliminary version of this paper appears in Proceedings of the 28th International Colloquium on Automata, Languages and Programming, Crete, Greece, 2001. This work was supported in part by the European Union under IST FET Project AEOLUS.

A. Ferreira is currently on leave as Head of Science Operations at the COST Office, Brussels, http://www.cost.esf.org.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Caragiannis, I., Ferreira, A., Kaklamanis, C. et al. Fractional Path Coloring in Bounded Degree Trees with Applications. Algorithmica 58, 516–540 (2010). https://doi.org/10.1007/s00453-009-9278-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-009-9278-3

Keywords

Search

Navigation