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Random walks on the vertices of transportation polytopes with constant number of sources
 Proc. 14th Ann. ACMSIAM Symp. Disc. Alg. (Baltimore, MD) 330–339, ACM
, 2003
"... We consider the problem of uniformly sampling a vertex of a transportation polytope with m sources and n destinations, where m is a constant. We analyse a natural random walk on the edgevertex graph of the polytope. The analysis makes use of the multicommodity flow technique of Sinclair [30] toget ..."
Abstract

Cited by 12 (3 self)
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We consider the problem of uniformly sampling a vertex of a transportation polytope with m sources and n destinations, where m is a constant. We analyse a natural random walk on the edgevertex graph of the polytope. The analysis makes use of the multicommodity flow technique of Sinclair [30] together with ideas developed by Morris and Sinclair [24, 25] for the knapsack problem, and Cryan et al. [3] for contingency tables, to establish that the random walk approaches the uniform distribution in time n O(m2). 1
ON THE GRAPHCONNECTIVITY OF SKELETA OF CONVEX POLYTOPES
, 801
"... Abstract. Given a ddimensional convex polytope P and nonnegative integer k not exceeding d − 1, let Gk(P) denote the simple graph on the node set of kdimensional faces of P in which two such faces are adjacent if there exists a (k + 1)dimensional face of P which contains them both. The graph Gk(P ..."
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Abstract. Given a ddimensional convex polytope P and nonnegative integer k not exceeding d − 1, let Gk(P) denote the simple graph on the node set of kdimensional faces of P in which two such faces are adjacent if there exists a (k + 1)dimensional face of P which contains them both. The graph Gk(P) is isomorphic to the dual graph of the (d − k)dimensional skeleton of the normal fan of P. For fixed values of k and d, the largest integer m such that Gk(P) is mvertexconnected for all ddimensional polytopes P is determined. This result generalizes Balinski’s theorem on the onedimensional skeleton of a ddimensional convex polytope. 1.
Combinatorial Polytope Enumeration
, 2009
"... We describe a provably complete algorithm for the generation of a tight, possibly exact, superset of all combinatorially distinct simple nfacet polytopes in R d, along with their graphs, fvectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d−simplex. Ou ..."
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We describe a provably complete algorithm for the generation of a tight, possibly exact, superset of all combinatorially distinct simple nfacet polytopes in R d, along with their graphs, fvectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d−simplex. Our generator has implications for several outstanding problems in polytope theory, including conjectures about the number of distinct polytopes, the edge expansion of polytopal graphs, and the dstep conjecture. 1