Random walks, totally unimodular matrices, and a randomised dual simplex algorithm

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.

[1]  R. Tennant.  Algebra , 1941, Nature.

[2]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[3]  R. Acevedo,et al.  Research report , 1967, Revista odontologica de Puerto Rico.

[4]  Norman Zadeh,et al.  A bad network problem for the simplex method and other minimum cost flow algorithms , 1973, Math. Program..

[5]  W. H. Cunningham,et al.  Theoretical Properties of the Network Simplex Method , 1979, Math. Oper. Res..

[6]  Richard P. Stanley,et al.  Two Combinatorial Applications of the Aleksandrov-Fenchel Inequalities , 1981, J. Comb. Theory, Ser. A.

[7]  P. Flajolet.  On approximate counting , 1982 .

[8]  D. Aldous.  Random walks on finite groups and rapidly mixing markov chains , 1983 .

[9]  Andrei Z. Broder,et al.  How hard is it to marry at random? (On the approximation of the permanent) , 1986, STOC '86.

[10]  Éva Tardos,et al.  A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs , 1986, Oper. Res..

[11]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, WG.

[12]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[13]  Martin E. Dyer,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991, JACM.

[14]  Denis Naddef,et al.  The hirsch conjecture is true for (0, 1)-polytopes , 1989, Math. Program..

[15]  Andrei Z. Broder,et al.  Generating random spanning trees , 1989, 30th Annual Symposium on Foundations of Computer Science.

[16]  Miklós Simonovits,et al.  The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[17]  Gil Kalai.  The Diameter of Graphs of Convex Polytopes and f-Vector Theory , 1990, Applied Geometry And Discrete Mathematics.

[18]  David Aldous,et al.  The Random Walk Construction of Uniform Spanning Trees and Uniform Labelled Trees , 1990, SIAM J. Discret. Math..

[19]  M. Dyer.  Computing the volume of convex bodies : a case where randomness provably helps , 1991 .

[20]  L. Khachiyan,et al.  On the conductance of order Markov chains , 1991 .

[21]  Robert E. Tarjan,et al.  Efficiency of the Primal Network Simplex Algorithm for the Minimum-Cost Circulation Problem , 1991, Math. Oper. Res..

[22]  Tomás Feder,et al.  Balanced matroids , 1992, STOC '92.

[23]  Mark Jerrum,et al.  Polynomial-Time Approximation Algorithms for the Ising Model , 1990, SIAM J. Comput..