Independence and port oracles for matroids, with an application to computational learning theory Collette R. Coullard Lisa Hellerstein Article Received: 12 September 1994 DOI :
10.1007/BF01844845

Cite this article as: Coullard, C.R. & Hellerstein, L. Combinatorica (1996) 16: 189. doi:10.1007/BF01844845
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 Research partially funded by NSF PYI Grant No. DDM-91-96083.

Research partially funded by NSF Grant No CCR-92-10957.

References [1]

D. Angluin, L. Hellerstein , and

M. Karpinski : Learning read-once formulas with queries,

J. of the Association for Computing Machinery
40 (1993) 185–210.

Google Scholar [2]

N. Bshouty, T. Hancock, L. Hellerstein , and

M. Karpinski : An algorithm to learn read-once threshold formulas, and transformations between learning models,

Computational Complexity
4 (1994) 37–61.

CrossRef Google Scholar [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.

[4]

R. E. Bixby , and

W. H. Cunningham : Converting linear programs to network problems,

Mathematics of Operations Research
5 (1980) 321–357.

Google Scholar [5]

R. E. Bixby , and

D. K. Wagner : An almost linear time algorithm for graph realization,

Mathematics of Operations Research
13 (1988) 99–123.

Google Scholar [6]

T. H. Brylawski , and

D. Lucas : Uniquely representable combinatorial geometries,

Teorie Combinatorie (Proc. 1973 Internat. Colloq.) 83–104, Accademia nazionale dei Lincei, Rome, (1976).

Google Scholar [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.

[8]

L. Hellerstein , andC. Coullard : Learning binary matroid ports,Proceedings of the 5th Annual SIAM Symposium on Discrete Algorithms (1994) 328–335.

[9]

P. M. Jensen and

B. Korte : Complexity of matroid property algorithms,

SIAM Journal of Computation ,

11 (1) (1982) 184–190.

CrossRef Google Scholar [10]

J. Kahn : On the uniqueness of matroid representations over GF(4).

Bull. London Math Soc.
20 (1988) 5–10.

Google Scholar [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.

[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.

[13]

A. Lehman : A solution of the Shannon switching game,

Journal of the Society of Industrial and Applied Mathematics
12 :4 (1964) 687–725.

CrossRef Google Scholar [14]

J. G. Oxley :

Matroid Theory , Oxford University Press, New York, (1992).

Google Scholar [15]

J. G. Oxley, D. Vertigan , and

G. 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]

L. Pitt , and

L. Valiant : Computational limitations on learning from examples,

J. ACM
35 (1988) 965–984.

CrossRef Google Scholar [17]

V. Raghavan , andS. Schach : Learning switch configurations,Proceedings of Third Annual Workshop on Computational Learning Theory Morgan Kaufmann Publishers (1990) 38–51.

[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.

[19]

P. D. Seymour : The forbidden minors of binary clutters,

J. London Math. Soc. (2)

12 (1975) 356–360.

Google Scholar [20]

P. D. Seymour : A note on the production of matroid minors,

J. of Combinatorial Theory (B)
22 (1977) 289–295.

CrossRef Google Scholar [21]

P. D. Seymour : The matroids with the max-flow min-cut property,

J. of Combinatorial Theory (B)
23 (1977) 189–222.

CrossRef Google Scholar [22]

P. D. Seymour : Recognizing graphic matroids,

Combinatorica
1 (1981) 75–78.

Google Scholar [23]

K. Truemper :

Matroid Decomposition , Academic Press, San Diego, (1992).

Google Scholar [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]

D. J. A. Welsh :

Matroid Theory , Academic Press, London, (1976).

Google Scholar Authors and Affiliations Collette R. Coullard Lisa Hellerstein 1. Department of Industrial Engineering and Management Sciences Northwestern University Evanston USA 2. Department of Electrical Engineering and Computer Science Evanston USA