Combinatorica

, Volume 16, Issue 2, pp 189–208

Independence and port oracles for matroids, with an application to computational learning theory

  • Collette R. Coullard
  • Lisa Hellerstein
Article

Abstract

Given a matroidM with distinguished elemente, aport oracie with respect toe reports whether or not a given subset contains a circuit that containse. The first main result of this paper is an algorithm for computing ane-based ear decomposition (that is, an ear decomposition every circuit of which contains elemente) of a matroid using only a polynomial number of elementary operations and port oracle calls. In the case thatM is binary, the incidence vectors of the circuits in the ear decomposition form a matrix representation forM. Thus, this algorithm solves a problem in computational learning theory; it learns the class ofbinary matroid port (BMP) functions with membership queries in polynomial time. In this context, the algorithm generalizes results of Angluin, Hellerstein, and Karpinski [1], and Raghavan and Schach [17], who showed that certain subclasses of the BMP functions are learnable in polynomial time using membership queries. The second main result of this paper is an algorithm for testing independence of a given input set of the matroidM. This algorithm, which uses the ear decomposition algorithm as a subroutine, uses only a polynomial number of elementary operations and port oracle calls. The algorithm proves a constructive version of an early theorem of Lehman [13], which states that the port of a connected matroid uniquely determines the matroid.

Mathematics Subject Classification (1991)

05 B 35 68 T 05 68 Q 20 68 Q 25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Angluin, L. Hellerstein, andM. Karpinski: Learning read-once formulas with queries,J. of the Association for Computing Machinery 40 (1993) 185–210.Google Scholar
  2. [2]
    N. Bshouty, T. Hancock, L. Hellerstein, andM. Karpinski: An algorithm to learn read-once threshold formulas, and transformations between learning models,Computational Complexity 4 (1994) 37–61.CrossRefGoogle Scholar
  3. [3]
    R. E. Bixby: Matroids and operations research. in:Advanced Techniques in the Practice of Operations Research, (H. J. Greenberg, F. H. Murphy, and S. H. Shaw, eds.), North-Holland Publishers (1980) 333–459.Google Scholar
  4. [4]
    R. E. Bixby, andW. H. Cunningham: Converting linear programs to network problems,Mathematics of Operations Research 5 (1980) 321–357.Google Scholar
  5. [5]
    R. E. Bixby, andD. K. Wagner: An almost linear time algorithm for graph realization,Mathematics of Operations Research 13 (1988) 99–123.Google Scholar
  6. [6]
    T. H. Brylawski, andD. Lucas: Uniquely representable combinatorial geometries,Teorie Combinatorie (Proc. 1973 Internat. Colloq.) 83–104, Accademia nazionale dei Lincei, Rome, (1976).Google Scholar
  7. [7]
    D. Hausmann, andB. Korte: The relative strength of oracles for independence systems, in:Special Topics of Applied Mathematics, (J. Frehse, D. Pallaschke, and U Trottenberg, eds.), North-Holland Publishers (1980) 195–211.Google Scholar
  8. [8]
    L. Hellerstein, andC. Coullard: Learning binary matroid ports,Proceedings of the 5th Annual SIAM Symposium on Discrete Algorithms (1994) 328–335.Google Scholar
  9. [9]
    P. M. Jensen andB. Korte: Complexity of matroid property algorithms,SIAM Journal of Computation,11 (1) (1982) 184–190.CrossRefGoogle Scholar
  10. [10]
    J. Kahn: On the uniqueness of matroid representations over GF(4).Bull. London Math Soc. 20 (1988) 5–10.Google Scholar
  11. [11]
    M. Kearns, M. Li, L. Pitt, andL. Valiant: On the learnability of boolean formulae,Proc. 19th ACM Symposium on Theory of Computing (1987) 285–295.Google Scholar
  12. [12]
    M. Kearns, andL. Valiant: Cryptographic limitations on learning boolean formulae and finite automata,Proc. 21st ACM Symposium on Theory of Computing (1989) 433–444.Google Scholar
  13. [13]
    A. Lehman: A solution of the Shannon switching game,Journal of the Society of Industrial and Applied Mathematics 12:4 (1964) 687–725.CrossRefGoogle Scholar
  14. [14]
    J. G. Oxley:Matroid Theory, Oxford University Press, New York, (1992).Google Scholar
  15. [15]
    J. G. Oxley, D. Vertigan, andG. Whittle: On inequivalent representations of matroids over finite fields, Technical Report, Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, (1994).Google Scholar
  16. [16]
    L. Pitt, andL. Valiant: Computational limitations on learning from examples,J. ACM 35 (1988) 965–984.CrossRefGoogle Scholar
  17. [17]
    V. Raghavan, andS. Schach: Learning switch configurations,Proceedings of Third Annual Workshop on Computational Learning Theory Morgan Kaufmann Publishers (1990) 38–51.Google Scholar
  18. [18]
    V. Raghavan, andD. Wilkins: Learning μ-branching programs with queries,Proceedings of the Sixth Annual Workshop on Computational Learning Theory, ACM Press (1993) 27–36.Google Scholar
  19. [19]
    P. D. Seymour: The forbidden minors of binary clutters,J. London Math. Soc. (2)12 (1975) 356–360.Google Scholar
  20. [20]
    P. D. Seymour: A note on the production of matroid minors,J. of Combinatorial Theory (B) 22 (1977) 289–295.CrossRefGoogle Scholar
  21. [21]
    P. D. Seymour: The matroids with the max-flow min-cut property,J. of Combinatorial Theory (B) 23 (1977) 189–222.CrossRefGoogle Scholar
  22. [22]
    P. D. Seymour: Recognizing graphic matroids,Combinatorica 1 (1981) 75–78.Google Scholar
  23. [23]
    K. Truemper:Matroid Decomposition, Academic Press, San Diego, (1992).Google Scholar
  24. [24]
    W. T. Tutte: An algorithm for determining whether a given binary matroid is graphic,Proc. Amer. Math. Soc. 11 (1960) 905–917.Google Scholar
  25. [25]
    D. J. A. Welsh:Matroid Theory, Academic Press, London, (1976).Google Scholar

Copyright information

© Akadémiai Kiadó 1996

Authors and Affiliations

  • Collette R. Coullard
    • 1
  • Lisa Hellerstein
    • 2
  1. 1.Department of Industrial Engineering and Management SciencesNorthwestern UniversityEvanstonUSA
  2. 2.Department of Electrical Engineering and Computer ScienceEvanstonUSA

Personalised recommendations