In this paper, we develop a new “robust mixing” framework for reasoning about adversarially modified Markov Chains (AMMC). Let ℙ be the transition matrix of an irreducible Markov Chain with stationary distribution π. An adversary announces a sequence of stochastic matrices \(\{{\mathbb{A}}_t\}_{t > 0}\) satisfying \(\pi{\mathbb{A}}_t = \pi\). An AMMC process involves an application of ℙ followed by \({\mathbb{A}}_t\) at time t. The robust mixing time of an irreducible Markov Chain ℙ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains.

Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time t, exchanges the card at position \(t {\pmod n}\) with a random card. Mossel, Peres and Sinclair (2004) showed that the mixing time of this process lies between (0.0345+o(1))nlogn and Cnlogn + O(n) (with C ≈4 ×105). We reduce the constant C to 1 by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time ≤nlogn + O(n) when the adversarial strategies are limited to those which preserve the symmetry of the underlying Markov Chain.

Reversible liftings: Chen, Lovász and Pak showed that for a reversible ergodic Markov Chain ℙ, any reversible lifting ℚ of ℙ must satisfy \({\mathcal{T}}({\mathbb{P}}) \leq {\mathcal{T}}(\mathbb{Q})\log (1/\pi_*)\) where π * is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that \({\mathcal{T}}(\mathbb{Q}) \geq r({\mathbb{P}})\) where r(ℙ) is the relaxation time of ℙ. This helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.


Markov Chain Stationary Distribution Convex Combination Transition Probability Matrix Stochastic Matrice 
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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Murali K. Ganapathy
    • 1
  1. 1.University of ChicagoChicagoUSA

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