## Abstract

We discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix. We also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron.

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## References

D. Aldous, “Random walks on finite groups and rapidly mixing Markov chains,”

*Séminaire de Probabilités XVIII, 1981–82, Springer Lecture Note in Mathematics No. 986*(Springer, Berlin, 1982) pp. 243–297.D. Aldous, “The random walk construction of uniform spanning trees and uniform labelled trees,”

*SIAM Journal on Discrete Mathematics*3 (1990) 450–465.D. Applegate and R. Kannan, “Sampling and integration of near logconcave functions,”

*Proceedings of the 23rd ACM Symposium on Theory of Computing*(1991) pp. 156–163.J.W. Archbold,

*Algebra*(Pitman, London, 1958).A.Z. Broder, “How hard is it to marry at random? (On the approximation of the permanent),”

*Proceedings of the 18th ACM Symposium on Theory of Computing*(1986) pp. 50–58.A.Z. Broder, “Generating random spanning trees,”

*Proceedings of the 30th Annual Symposium on Foundations of Computer Science*(1989) pp. 442–447.Y.D. Burago and V.A. Zalgaller,

*Geometric Inequalities*(Springer, Berlin, 1980).W.H. Cunningham, “Theoretical properties of the network simplex method,”

*Mathematics of Operations Research*4, 196–208.M.E. Dyer, “Approximation of mixed volumes,” in preparation.

M.E. Dyer and A.M. Frieze, “Computing the volume of convex bodies: a case where randomness probably helps,” to appear in: B. Bollobás, ed.,

*Proceedings of AMS Short Course on Probability and Combinatorics.*M.E. Dyer, A.M. Frieze and R. Kannan, “A random polynomial time algorithm for approximating the volume of convex bodies,”

*Journal of the ACM*38 (1991), 1–17.T. Feder and M. Mihail, “Balanced matroids,”

*Proceedings of the 24th Annual ACM Symposium on Theory of Computing*(1992) pp. 26–38.M.R. Jerrum and A.J. Sinclair, “Approximating the permanent,”

*SIAM Journal on Computing*18 (1989) 1149–1178.M.R. Jerrum and A.J. Sinclair, “Polynomial-time approximation algorithms for the Ising model,” Department of Computer Science, Edinburgh University (Edinburgh, 1989).

G. Kalai, “The diameter of graphs of convex polytopes and

*f*-vector theory,” to appear in:*DIMACS Series in Discrete Mathematics and Theoretical Computer Science.*A. Karzanov and L.G. Khachyan, “On the conductance of order Markov chains,” Technical Report DCS TR 268, Rutgers University (New Brunswick, NJ, 1990).

L. Lovász and M. Simonovits, “The mixing rate of Markov chains, an isoperimetric inequality and computing the volume,”

*Proceedings of the 31st Annual Symposium on Foundations of Computer Science*(1990) pp. 364–355.N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, Equation of state calculation by fast computing machines,

*Journal of Chemical Physics*21 (1953) 1087–1091.M. Mihail and P. Winkler, “On the number of Euler orientations of a graph,’

*Proceedings of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms*(1992) pp. 138–145.D. Naddef, “The Hirsch conjecture is true for 0–1 polytopes,”

*Mathematical Programming*45 (1989) 109–110.A. Schrijver,

*The Theory of Linear and Integer Programming*(Wiley, Chichester, 1986).A.J. Sinclair and M.R. Jerrum, “Approximate counting, uniform generation and rapidly mixing Markov chains,”

*Information and Computation*82 (1989) 93–133.R.M. Stanley, “Two combinatorial applications of the Aleksandrov-Fenchel inequalities,”

*Journal of Combinatorial Theory A*31 (1981) 56–65.E. Tardos, “A strongly polynomial algorithm to solve combinatorial linear programs,”

*Operations Research*34 (1986) 250–256.R.E. Tarjan, “Efficiency of the primal network simplex method algorithm for the minimum-cost circulation problem,”

*Mathematics of Operations Research*16 (1991), 272–291.N. Zadeh, “A bad network for the simplex method and other minimum cost flow algorithms,”

*Mathematical Programming*5 (1973) 255–266.

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Supported by NATO grant RG0088/89.

Corresponding author. Supported by NSF grants CCR-8900112, CCR-9024935 and NATO grant RG0088/89.

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Dyer, M., Frieze, A. Random walks, totally unimodular matrices, and a randomised dual simplex algorithm.
*Mathematical Programming* **64**, 1–16 (1994). https://doi.org/10.1007/BF01582563

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DOI: https://doi.org/10.1007/BF01582563