Independence and port oracles for matroids, with an application to computational learning theory
 Collette R. Coullard,
 Lisa Hellerstein
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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 anebased 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.
 Angluin, D., Hellerstein, L., Karpinski, M. (1993) Learning readonce formulas with queries. J. of the Association for Computing Machinery 40: pp. 185210
 Bshouty, N., Hancock, T., Hellerstein, L., Karpinski, M. (1994) An algorithm to learn readonce threshold formulas, and transformations between learning models. Computational Complexity 4: pp. 3761 CrossRef
 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.), NorthHolland Publishers (1980) 333–459.
 Bixby, R. E., Cunningham, W. H. (1980) Converting linear programs to network problems. Mathematics of Operations Research 5: pp. 321357
 Bixby, R. E., Wagner, D. K. (1988) An almost linear time algorithm for graph realization. Mathematics of Operations Research 13: pp. 99123
 Brylawski, T. H., Lucas, D. (1976) Uniquely representable combinatorial geometries. Teorie Combinatorie. Accademia nazionale dei Lincei, Rome, pp. 83104
 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.), NorthHolland Publishers (1980) 195–211.
 L. Hellerstein, andC. Coullard: Learning binary matroid ports,Proceedings of the 5th Annual SIAM Symposium on Discrete Algorithms (1994) 328–335.
 Jensen, P. M., Korte, B. (1982) Complexity of matroid property algorithms. SIAM Journal of Computation 11: pp. 184190 CrossRef
 Kahn, J. (1988) On the uniqueness of matroid representations over GF(4). Bull. London Math Soc. 20: pp. 510
 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.
 M. Kearns, andL. Valiant: Cryptographic limitations on learning boolean formulae and finite automata,Proc. 21st ACM Symposium on Theory of Computing (1989) 433–444.
 Lehman, A. (1964) A solution of the Shannon switching game. Journal of the Society of Industrial and Applied Mathematics 12: pp. 687725 CrossRef
 Oxley, J. G. (1992) Matroid Theory. Oxford University Press, New York
 Oxley, J. G., Vertigan, D., Whittle, G. (1994) On inequivalent representations of matroids over finite fields. Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803
 Pitt, L., Valiant, L. (1988) Computational limitations on learning from examples. J. ACM 35: pp. 965984 CrossRef
 V. Raghavan, andS. Schach: Learning switch configurations,Proceedings of Third Annual Workshop on Computational Learning Theory Morgan Kaufmann Publishers (1990) 38–51.
 V. Raghavan, andD. Wilkins: Learning μbranching programs with queries,Proceedings of the Sixth Annual Workshop on Computational Learning Theory, ACM Press (1993) 27–36.
 Seymour, P. D. (1975) The forbidden minors of binary clutters. J. London Math. Soc. 12: pp. 356360
 Seymour, P. D. (1977) A note on the production of matroid minors. J. of Combinatorial Theory (B) 22: pp. 289295 CrossRef
 Seymour, P. D. (1977) The matroids with the maxflow mincut property. J. of Combinatorial Theory (B) 23: pp. 189222 CrossRef
 Seymour, P. D. (1981) Recognizing graphic matroids. Combinatorica 1: pp. 7578
 Truemper, K. (1992) Matroid Decomposition. Academic Press, San Diego
 Tutte, W. T. (1960) An algorithm for determining whether a given binary matroid is graphic. Proc. Amer. Math. Soc. 11: pp. 905917
 Welsh, D. J. A. (1976) Matroid Theory. Academic Press, London
 Title
 Independence and port oracles for matroids, with an application to computational learning theory
 Journal

Combinatorica
Volume 16, Issue 2 , pp 189208
 Cover Date
 19960601
 DOI
 10.1007/BF01844845
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 05 B 35
 68 T 05
 68 Q 20
 68 Q 25
 Industry Sectors
 Authors

 Collette R. Coullard ^{(1)}
 Lisa Hellerstein ^{(2)}
 Author Affiliations

 1. Department of Industrial Engineering and Management Sciences, Northwestern University, 60208, Evanston, IL, USA
 2. Department of Electrical Engineering and Computer Science, 60208, Evanston, IL, USA