M. Dyer, A. Frieze, R. Kannan, A. Kapoor, L. Perkovic and U. Vazirani. A sub-exponential time algorithm for approximating the number of solutions to a multidimensional knapsack problem. Combinatorics, Probability and Computing, 2 (1993), pp. 271-284.  Home/Search   Document Details and Download   Summary   Related Articles

.... to analyze a variety of Markov chains including those for sampling perfect matchings and approximating the permanent [20, 8] estimating the partition function of the Ising model [21] sampling bases of balanced matroids [17] sampling regular bipartite graphs [25] sampling 0 1 knapsack solutions [12], etc. All these papers with the exception of [17] use more or less the same technique to bound the path congestion that is due to [20] they use the state space to somehow encode the paths that use any given transition, so that the number of paths through any edge will be comparable to the ....

....with uniform stationary distribution. Despite all the recent activity in proving rapid mixing, this simple example was not known to be rapidly mixing until the work of [34] The best prior known bound on the mixing time, obtained via the canonical paths technique, was exp(O( p n(log n) 5=2 ) [12], which beats the trivial bound of exp(O(n) but is still exponential. We will now sketch the proof of [34] that this chain has a mixing time of O(n 8 ) and is thus indeed rapidly mixing. The proof will follow the resistance approach, i.e. we will find a flow f that routes one unit of flow ....

M. Dyer, A. Frieze, R. Kannan, A. Kapoor, L. Perkovic and U. Vazirani. A sub-exponential time algorithm for approximating the number of solutions to a multidimensional knapsack problem. Combinatorics, Probability and Computing, 2 (1993), pp. 271-284.

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