Random generation of combinatorial structures from a uniform
Theoretical Computer Science
Conductance and the rapid mixing property for Markov chains: the approximation of permanent resolved
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Basis pair graphs of transversal matroids are connected
Discrete Mathematics
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
On the problem of approximating the number of bases of a matroid
Information Processing Letters
Algorithms for random generation and counting: a Markov chain approach
Algorithms for random generation and counting: a Markov chain approach
Generating random spanning trees
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Markov chains and polynomial time algorithms
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
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We consider the expansion property for the bases-exchange graph of a matroid, which gives an effective randomized approximation scheme for the number of bases. Mihail and Feder in [6] proved that this property holds for the class of so-called balanced matroids, including all regular matroids. We present a much simpler proof of similar result for all matroids satisfying exchange-preservation property, defined as some graph-theoretic condition on the family of internal exchange graphs. We apply a method of bounding path congestion proposed in [8], which is of additional interest as it was suggested in [6] that this technique seems to be unsuitable to matroid related graphs. As an illustrating example we apply our results to a subclass of transversal matroids.