, Volume 58, Issue 2, pp 516–540 | Cite as

Fractional Path Coloring in Bounded Degree Trees with Applications

  • I. Caragiannis
  • A. Ferreira
  • C. Kaklamanis
  • S. Pérennes
  • H. RivanoEmail author


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.


Fractional coloring Path coloring Linear relaxation Approximation algorithms Wavelength division multiplexing Optical networks Fixed parameter tractable problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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. 2.
    Azuma, K.: Weighted sum of certain dependent random variables. Tohoku Math. J. 19, 357–367 (1967) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 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. 4.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics Series, vol. 244. Springer, Berlin (2008) zbMATHGoogle Scholar
  5. 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. 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. 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) CrossRefGoogle Scholar
  8. 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) CrossRefGoogle Scholar
  9. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Feige, U., Kilian, J.: Zero knowledge and the chromatic number. J. Comput. Syst. Sci. 57(2), 187–199 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 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 Google Scholar
  12. 12.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  13. 13.
    Garg, N.: Multicommodity flows and approximation algorithms. Ph.D. thesis, Indian Institute of Technology, Delhi, April (1994) Google Scholar
  14. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 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. 16.
    Golumbic, M.C., Jamison, R.E.: The edge intersection graphs of paths in a tree. J. Comb. Theory 38, 8–22 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2, 2nd corrected edn. Springer, Berlin (1993) zbMATHGoogle Scholar
  19. 19.
    Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problems. PWS-Kent, Boston (1997) Google Scholar
  20. 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. 21.
    Karapetyan, I.A.: On coloring of arc graphs. Dokl. Akad. Nauk Armianskoi CCP 70(5), 306–311 (1980). In Russian zbMATHMathSciNetGoogle Scholar
  22. 22.
    König, D.: Graphok és matrixok. Mat. Fiz. Lapok 116–119 (1931) Google Scholar
  23. 23.
    Kumar, V.: Approximating arc circular colouring and bandwidth allocation in all-optical ring networks. In: APPROX’98 (1998) Google Scholar
  24. 24.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Niessen, T., Kind, J.: The round-up property of the fractional chromatic number for proper circular arc graphs, J. Graph Theory (1998) Google Scholar
  26. 26.
    Tarjan, R.E.: Decomposition by clique separators. Discrete Math. 55(2), 221–232 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Tucker, A.: Coloring a family of circular arcs. SIAM J. Appl. Math. 29(3), 493–502 (1975) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • I. Caragiannis
    • 1
  • A. Ferreira
    • 2
  • C. Kaklamanis
    • 1
  • S. Pérennes
    • 2
  • H. Rivano
    • 2
    Email author
  1. 1.Computer Technology InstitutePatrasGreece
  2. 2.MASCOTTE Project, CNRS-I3S-INRIASophia AntipolisFrance

Personalised recommendations