Abstract
We present algorithms for thek-Matroid Intersection Problem and for the Matroidk-Parity Problem when the matroids are represented over the field of rational numbers andk > 2. The computational complexity of the algorithms is linear in the cardinality and singly exponential in the rank of the matroids. As an application, we describe new polynomially solvable cases of thek-Dimensional Assignment Problem and of thek-Dimensional Matching Problem. The algorithms use some new identities in multilinear algebra including the generalized Binet—Cauchy formula and its analogue for the Pfaffian. These techniques extend known methods developed earlier fork = 2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A.V. Aho, J.E. Hopcroft and J.D. Ullman,The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, MA, 1974).
A.I. Barvinok, “Optimization problems on matroids and exponential sums,” in: E. Balas, G. Cornuéjols and R. Kannan, eds.,Integer Programming and Combinatorial Optimization, Proceedings of the Second IPCO Conference, Carnegie Mellon University (1992) pp. 316–333.
P.M. Camerini, G. Galbiati and F. Maffioli, “Random pseudo-polynomial algorithms for exact matroid problems,”Journal of Algorithms 13 (1992) 258–273.
A. Cayley, “On the theory of determinants,” in:Collected Papers, Vol. 1 (Cambridge University Press, Cambridge, 1889) pp. 63–80.
A. Cayley, “On the theory of linear transformations,” in:Collected Papers, Vol. 1 (Cambridge University Press, Cambridge, 1889) pp. 80–94.
G. Galbiati and F. Maffioli, “On the computation of pfaffians,”Discrete Applied Mathematics 51 (1994) 269–275.
M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-completeness (Freeman, San Francisco, CA, 1979).
F. Harary,Graph Theory (Addison-Wesley, Reading, MA, 1969).
E.L. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976).
L. Lovász, “On determinants, matchings and random algorithms,” in: L. Budach, ed.,Fundamentals of Computation Theory, Proceedings of the Conference on Algebraic, Arithmetic and Categorial Methods in Computation Theory (Akademie-Verlag, Berlin, 1979) pp. 565–574.
L. Lovász, “The matroid matching problem,” in: L. Lovász and V.T. Sós, eds.,Algebraic Methods in Graph Theory (North-Holland, Amsterdam, 1981) pp. 495–517.
L. Lovász and M.D. Plummer,Matching Theory (Akadémia Kiadò, Budapest/North-Holland, Amsterdam, 1986).
I. Satake,Linear Algebra (Marcel Dekker, New York, 1975).
Yu.G. Smetanin and L.G. Khachiyan, “Use of pseudopolynomial algorithms for some problems of combinatorial optimization with constraints,”Izvestiya Akademii Nauk SSSR Tekhnicheskaya Kibernetika (6) (1986) 139–144 (in Russian); translated in:Soviet Journal of Computer and Systems Science 25 (2) (1987) 161–165.
N.P. Sokolov,Spatial Matrices and their Applications (Fizmatgiz, Moscow, 1960, in Russian).
D.J.A. Welsh,Matroid Theory (Academic Press, London, 1976).
Author information
Authors and Affiliations
Additional information
A preliminary version of this paper appeared in the Proceedings of the Second IPCO Conference [2].
Supported by the Mittag-Leffler Institute and KTH, Stockholm.
Rights and permissions
About this article
Cite this article
Barvinok, A.I. New algorithms for lineark-matroid intersection and matroidk-parity problems. Mathematical Programming 69, 449–470 (1995). https://doi.org/10.1007/BF01585571
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585571