Skip to main content

Implicit Computation of Maximum Bipartite Matchings by Sublinear Functional Operations

  • Conference paper
Theory and Applications of Models of Computation (TAMC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7287))

  • 1112 Accesses

Abstract

The maximum bipartite matching problem, an important problem in combinatorial optimization, has been studied for a long time. In order to solve problems for very large structured graphs in reasonable time and space, implicit algorithms have been investigated. Any object to be manipulated is binary encoded and problems have to be solved mainly by functional operations on the corresponding Boolean functions. OBDDs are a popular data structure for Boolean functions, therefore, OBDD-based algorithms have been used as an heuristic approach to handle large input graphs. Here, two OBDD-based maximum bipartite matching algorithms are presented, which are the first ones using only a sublinear number of operations (with respect to the number of vertices of the input graph) for a problem unknown to be in NC, the complexity class that contains all problems computable in deterministic polylogarithmic time with polynomially many processors. Furthermore, the algorithms are experimentally evaluated.

The first two authors have been supported by DFG project BO 2755/1-1.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)

    MATH  Google Scholar 

  2. Berge, C.: Two theorems in graph theory. Proc. of National Academy of Science of the USA 43(9), 842–844 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bloem, R., Gabow, H.N., Somenzi, F.: An Algorithm for Strongly Connected Component Analysis in n logn Symbolic Steps. In: Johnson, S.D., Hunt Jr., W.A. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 37–54. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  4. Bollig, B.: Exponential space complexity for OBDD-based reachability analysis. Information Processing Letters 110, 924–927 (2010)

    Article  MathSciNet  Google Scholar 

  5. Bollig, B.: Exponential Space Complexity for Symbolic Maximum Flow Algorithms in 0-1 Networks. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 186–197. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  6. Bollig, B.: On Symbolic OBDD-Based Algorithms for the Minimum Spanning Tree Problem. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part II. LNCS, vol. 6509, pp. 16–30. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  7. Bollig, B.: On Symbolic Representations of Maximum Matchings and (Un)directed Graphs. In: Calude, C.S., Sassone, V. (eds.) TCS 2010. IFIP AICT, vol. 323, pp. 286–300. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  8. Bollig, B., Löbbing, M., Wegener, I.: On the effect of local changes in the variable ordering of ordered decision diagrams. Information Processing Letters 59, 233–239 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bollig, B., Pröger, T.: An Efficient Implicit OBDD-Based Algorithm for Maximal Matchings. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 143–154. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  10. Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. on Computers 35, 677–691 (1986)

    Article  MATH  Google Scholar 

  11. Charles, D.X., Chickering, M., Devanur, N.R., Jain, K., Sanghi, M.: Fast algorithms for finding matchings in lopsided bipartite graphs with applications to display ads. In: Proc. of ACM Conference on Electronic Commerce 2010, pp. 121–128 (2010)

    Google Scholar 

  12. Cherkassky, B.V., Goldberg, A.V., Martin, P., Setubal, J.C., Stolfi, J.: Augment or push: a computational study of bipartite matching and unit-capacity flow algorithms. ACM Journal of Experimental Algorithmics 3, 8 (1998)

    Article  MathSciNet  Google Scholar 

  13. Feigenbaum, J., Kannan, S., Vardi, M.V., Viswanathan, M.: Complexity of Problems on Graphs Represented as OBDDs. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 216–226. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  14. Gentilini, R., Piazza, C., Policriti, A.: Computing strongly connected components in a linear number of symbolic steps. In: Proc. of SODA, pp. 573–582. ACM Press (2003)

    Google Scholar 

  15. Gentilini, R., Piazza, C., Policriti, A.: Symbolic graphs: linear solutions to connectivity related problems. Algorithmica 50, 120–158 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goldberg, A.V., Plotkin, S.A., Shmoys, D.B., Tardos, E.: Using interior-point methods for fast parallel algorithms for bipartite matching and related problems. SIAM Journal on Computing 21(1), 140–150 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goldberg, A.V., Plotkin, S.K., Vaidya, P.M.: Sublinear time parallel algorithms for matching and related problems. Journal of Algorithms 14(2), 180–213 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hachtel, G.D., Somenzi, F.: A symbolic algorithm for maximum flow in 0-1 networks. Formal Methods in System Design 10, 207–219 (1997)

    Article  Google Scholar 

  19. Hopcroft, J.E., Karp, R.M.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing 2(4), 225–231 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  20. Iwano, K.: An improvement of Goldberg, Plotkin, and Vaidya’s maximal node-disjoint paths algorithm. Information Processing Letters 32, 25–27 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Monien, B., Preis, R., Diekmann, R.: Quality matching and local improvement for multilevel graph-partitioning. Parallel Computing 26, 1609–1634 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Möhring, R.H., Müller-Hannemann, M.: Complexity and modeling aspects of mesh refinement into quadrilaterals. Algorithmica 26, 148–172 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Negruseri, C.S., Pasoi, M.B., Stanley, B., Stein, C., Strat, C.G.: Solving maximum flow problems on real world bipartite graphs. In: Proc. of ALENEX, pp. 14–28. SIAM (2009)

    Google Scholar 

  24. Sawitzki, D.: Implicit Flow Maximization by Iterative Squaring. In: Van Emde Boas, P., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2004. LNCS, vol. 2932, pp. 301–313. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  25. Sawitzki, D.: Exponential Lower Bounds on the Space Complexity of OBDD-Based Graph Algorithms. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 781–792. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  26. Sawitzki, D.: The Complexity of Problems on Implicitly Represented Inputs. In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2006. LNCS, vol. 3831, pp. 471–482. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  27. Sawitzki, D.: Implicit simulation of FNC algorithms. ECCC Report TR07-028 (2007)

    Google Scholar 

  28. Spencer, T.H.: Parallel Approximate Matching. Parallel Algorithms and Applications 2(1-2), 115–121 (1994)

    Article  MATH  Google Scholar 

  29. Wegener, I.: Branching Programs and Binary Decision Diagrams - Theory and Applications. SIAM Monographs on Discrete Mathematics and Applications (2000)

    Google Scholar 

  30. Woelfel, P.: Symbolic topological sorting with OBDDs. Journal of Discrete Algorithms 4(1), 51–71 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bollig, B., Gillé, M., Pröger, T. (2012). Implicit Computation of Maximum Bipartite Matchings by Sublinear Functional Operations. In: Agrawal, M., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2012. Lecture Notes in Computer Science, vol 7287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29952-0_45

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-29952-0_45

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29951-3

  • Online ISBN: 978-3-642-29952-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics