Algorithmica

, 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. Rivano
Article
  • 96 Downloads

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.

Keywords

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

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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
  1. 1.Computer Technology InstitutePatrasGreece
  2. 2.MASCOTTE Project, CNRS-I3S-INRIASophia AntipolisFrance

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