Advertisement

Approximating minimum weight perfect matchings for complete graphs satisfying the triangle inequality

  • N. W. Holloway
  • S. Ravindran
  • A. M. Gibbons
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 790)

Abstract

We describe an O(log3n) time NC approximation algorithm for the CREW P-RAM, using n3/ log n processors with a 2log3n performance ratio, for the problem of finding a minimum-weight perfect matching in complete graphs satisfying the triangle inequality. The algorithm is conceptually very simple and has a work measure within a factor of log2n of the best exact sequential algorithm. This is the first NC approximation algorithm for the problem with a sub-linear performance ratio. As was the case in the development of sequential complexity theory, matching problems are on the boundary of what problems might ultimately be described as tractable for parallel computation. Future work in this area is likely to decide whether these ought to be regarded as those problems in NC or those problems in RNC.

keywords

approximation algorithms parallel algorithms matching 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Dahlhaus and M. Karpinski, “Parallel Construction of Perfect Matching and Hamiltonian Cycles on Dense Graphs”, Theoretical computer science 61 (1988) 121–136Google Scholar
  2. 2.
    J. Edmonds, “Matching and Polyhedrons with 0,1 Vertices”, Journal of Research of the National Bureau of Standards B, 125–130 (1965).Google Scholar
  3. 3.
    J. Edmonds, “Paths, trees and flowers”, Canadian Journal of Mathematics 17 (1965) 449–67.Google Scholar
  4. 4.
    H.N. Gabow, “Implementations of Algorithm for Maximum Matching on Nonbiparitte Graphs”, Ph.D. Dissertation, Dept. of Computer Science, Stanford University, 1974.Google Scholar
  5. 5.
    D.Y. Grigoriev and M. Karpinski, “The Matching Problem for Bipartite Graphs with Polynomially Bounded Permanents is in NC”, Proceedings of the Annual IEEE Symposium on Foundations of Computer Science (1987), 166–172.Google Scholar
  6. 6.
    Z. Galil and V. Pan, “Improved Processor Bounds for Combinatorial Problems in RNC”, Combinatorica 8 (1988) 189–200.Google Scholar
  7. 7.
    A.M. Gibbons, Algorithmic Graph Theory, Cambridge University Press (1985).Google Scholar
  8. 8.
    A.M. Gibbons and W. Rytter, Efficient Parallel Algorithms, Cambridge University Press (1988).Google Scholar
  9. 9.
    A.M. Gibbons and P.G. Spirakis (eds), Lectures on Parallel Computation, Cambridge University Press (1993).Google Scholar
  10. 10.
    D. Hembold and E. Mayer, “Two-processor Scheduling is in NC”, VLSI Algorithm and Architectures, editors: Makedon et al., Lecture Notes in Computer Science, Vol. 227 (1986) 12–25.Google Scholar
  11. 11.
    M. Iri, K. Murota and S. Matsui, “Linear-time Approximation Algorithms for Finding the Minimum-Weight Perfect Matching on a Plane”, Information Processing Letters 12 (1981) 206–209.Google Scholar
  12. 12.
    A. Israeli and Y. Shiloach, “An improved algorithm for maximal matching”, Information Processing Letters” 33 (1986) 57–60.Google Scholar
  13. 13.
    D.B. Johnson and P. Metaxas, “Connected components in O(log3/2 ¦V¦) parallel time for the CREW PRAM”, FOCS, 1991Google Scholar
  14. 14.
    D.R. Karger, N. Nisan and M. Parnas, “Fast Connected Components Algorithms for the EREW PRAM”, 4th Annual ACM Symposium on Parallel Algorithms and Architectures, 373–381 (1992)Google Scholar
  15. 15.
    R. Karp and V. Ramachandran, “Parallel algorithms for Shared Memory Machines”, Handbook of Theoretical Computer Science, J. van Leeuwen (editor), vol.1, Elsevier and MIT Press (1991).Google Scholar
  16. 16.
    R. Karp, E. Upfal and A. Wigderson, “Constructing a Perfect Matching is in Random NC”, Proceedings of the Annual ACM Symposium on Theory of Computing (1985), 22–32.Google Scholar
  17. 17.
    L. Kucera, “Parallel computation and conflicts in memory access”, Information Processing Letters, Vol. 14, (1982) 93–96Google Scholar
  18. 18.
    E.L. Lawler, Combinatorial Optimization: Networks and Matroids, Holt-Rinehart-Winston, New York, 1976.Google Scholar
  19. 19.
    F. Thomson Leighton, Introduction to Parallel Algorithms and Achitectures: Arrays • Trees • Hypercubes, Morgan Kaufmann, California (1992).Google Scholar
  20. 20.
    K. Mulmuley, U. Vazirani and V. Vazirani, “Matching is as Easy as Matrix Inverson”, Proceedings of the Annual ACM Symposium on Theory of Computing (1987), 345–354.Google Scholar
  21. 21.
    J. Naor, “Computing a Perfect Matching in a Line Graph”, Proceedings of the 9 th Conference on the Foundations of Software Technology and Theoretical Computer Science, (1989) 139–148.Google Scholar
  22. 22.
    C.N.K. Osiakwan and S.G. Akl, “The Maximum weight perfect matching problem for complete weighted graphs is in PC”, Proceedings of the 2nd IEEE Symposium on Parallel and Distributed Processing (1990) 880–887.Google Scholar
  23. 23.
    D.A. Plaisted, “Heuristic Matching for Graphs Satisfying the Triangle Inequality”, Journal of Algorithms 5 (1984) 163–179.Google Scholar
  24. 24.
    E.M. Reingold and R.E. Tarjan, “In a greedy Heuristic for Complete Matching”, SIAM Journal of Computing 10 (1981) 676–681.Google Scholar
  25. 25.
    Y. Shiloach and U. Vishkin, “An O(log n) parallel connectivity algorithm”, Journal of Algorithms, Vol. 3, (1982) 57–67.Google Scholar
  26. 26.
    K.J. Supowit, D.A. Plaisted and E.M. Reingold, “Heuristics for Weighted Perfect Matching”, Proceedings of the Annual ACM Symposium on Theory of Computing, (1980) 398–419.Google Scholar
  27. 27.
    R.E. Tarjan and U. Vishkin, “Finding biconnected components and computing tree functions in logarithmic parallel time”, Proceedings of the 25 th Annual IEEE Symposium on the Foundations of Computer Science, (1984), 12–20, also SIAM Journal of Computing, 14, 4(1985) 862–74.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • N. W. Holloway
    • 1
  • S. Ravindran
    • 1
  • A. M. Gibbons
    • 1
  1. 1.Department of Computer ScienceUniversity of WarwickCoventryUK

Personalised recommendations