L ∞ -Discrepancy Analysis of Polynomial-Time Deterministic Samplers Emulating Rapidly Mixing Chains
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- Shiraga T., Yamauchi Y., Kijima S., Yamashita M. (2014) L ∞ -Discrepancy Analysis of Polynomial-Time Deterministic Samplers Emulating Rapidly Mixing Chains. In: Cai Z., Zelikovsky A., Bourgeois A. (eds) Computing and Combinatorics. COCOON 2014. Lecture Notes in Computer Science, vol 8591. Springer, Cham
Markov chain Monte Carlo (MCMC) is a standard technique to sample from a target distribution by simulating Markov chains. In an analogous fashion to MCMC, this paper proposes a deterministic sampling algorithm based on deterministic random walk, such as the rotor-router model (a.k.a. Propp machine). For the algorithm, we give an upper bound of the point-wise distance (i.e., infinity norm) between the “distributions” of a deterministic random walk and its corresponding Markov chain in terms of the mixing time of the Markov chain. As a result, for uniform sampling of #P-complete problems, such as 0-1 knapsack solutions, linear extensions, matchings, etc., for which rapidly mixing chains are known, our deterministic algorithm provides samples with a “distribution” with a point-wise distance at most ε from the target distribution, in time polynomial in the input size and ε− 1.
Keywordsrotor-router model #P-complete Markov chain Monte Carlo mixing time
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