Randomized and Approximation Algorithms for Blue-Red Matching

  • Christos Nomikos
  • Aris Pagourtzis
  • Stathis Zachos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

We introduce the Blue-Red Matching problem: given a graph with red and blue edges, and a bound w, find a maximum matching consisting of at most w edges of each color. We show that Blue-Red Matching is at least as hard as the problem Exact Matching (Papadimitriou and Yannakakis, 1982), for which it is still open whether it can be solved in polynomial time. We present an RNC algorithm for this problem as well as two fast approximation algorithms. We finally show the applicability of our results to the problem of routing and assigning wavelengths to a maximum number of requests in all-optical rings.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Caragiannis, I.: Wavelength Management in WDM Rings to Maximize the Number of Connections. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 61–72. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Carlisle, M.C., Lloyd, E.L.: On the k-Coloring of Intervals. Discrete Applied Mathematics 59, 225–235 (1995)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Erlebach, T., Jansen, K.: The Complexity of Path Coloring and Call Scheduling. Theoretical Computer Science 255(1-2), 33–50 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Galbiati, G., Maffioli, F.: On the Computation of Pfaffians. Discrete Applied Mathematics 51(3), 269–275 (1994)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Garey, M., Johnson, D., Miller, G., Papadimitriou, C.: The Complexity of Coloring Circular Arcs and Chords. SIAM Journal on Algebraic Discrete Methods 1(2), 216–227 (1980)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Horowitz, E., Sahni, S.: On Computing the Exact Determinant of Matrices with Polynomial Entries. Journal of the ACM 22(1), 38–50 (1975)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Karzanov, A.V.: Maximum Matching of Given Weight in Complete and Complete Bipartite Graphs. Kibernetika 23(1), 7–11 (1987) (English translation in CYBNAW 23(1), 8–13 (1987))MathSciNetGoogle Scholar
  8. 8.
    Mahajan, M., Subramanya, P.R., Vinay, V.: The Combinatorial Approach Yields an NC Algorithm for Computing Pfaffians. Discrete Applied Mathematics 143(1-3), 1–16 (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Micali, S., Vazirani, V.V.: An O(n 2.5) Algorithm for Maximum Matching in General Graphs. In: Proceedings Twenty-first Annual Symposium on the Foundations of Computer Science, pp. 17–27 (1980)Google Scholar
  10. 10.
    Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as Easy as Matrix Inversion. Combinatorica 7(1), 105–113 (1987)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Nomikos, C., Pagourtzis, A., Zachos, S.: Satisfying a Maximum Number of Pre-Routed Requests in All-Optical Rings. Computer Networks 42(1), 55–63 (2003)MATHCrossRefGoogle Scholar
  12. 12.
    Nomikos, C., Pagourtzis, A., Zachos, S.: Minimizing Request Blocking in All-Optical Rings. In: Proceedings INFOCOM 2003, pp. 1771–1780 (2003)Google Scholar
  13. 13.
    Papadimitriou, C.H., Yannakakis, M.: The Complexity of Restricted Spanning Tree Problems. Journal of the ACM 29(2), 285–309 (1982)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Raghavan, P., Upfal, E.: Efficient Routing in All-Optical Networks. In: Proceedings of the 26th Annual ACM Symposium on the Theory of Computing STOC 1994, pp. 134–143. ACM Press, New York (1994)CrossRefGoogle Scholar
  15. 15.
    Stamoulis, G.: Maximum Matching Problems with Constraints (in Greek). Diploma Thesis, Department of Computer Science, University of Ioannina (2006)Google Scholar
  16. 16.
    Wan, P.-J., Liu, L.: Maximal Throughput in Wavelength-Routed Optical Networks. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 46, 15–26 (1998)MathSciNetGoogle Scholar
  17. 17.
    Yi, T., Murty, K.G., Spera, C.: Matchings in Colored Bipartite Networks. Discrete Applied Mathematics 121(1-3), 261–277 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Christos Nomikos
    • 1
  • Aris Pagourtzis
    • 2
  • Stathis Zachos
    • 2
    • 3
  1. 1.Department of Computer Science, University of Ioannina 
  2. 2.School of Electrical and Computer Engineering, National Technical University of Athens 
  3. 3.CIS Department, Brooklyn College, Cuny 

Personalised recommendations