Approximate Counting via Markov Chains

Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains

The paper studies effective approximate solutions to combinatorial counting and uniform generation problems. Using a technique based on the simulation of ergodic Markov chains, it is shown that, for self-reducible structures, almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polyn...

Approximate quantum Markov chains

This chapter reviews the concept of a Markov chain for random variables and then introduces a generalization to quantummechanical systems. We discuss the subtle differences between classical and quantumMarkov chains and point out difficulties that show up in the quantum case. Unlike the classical case that has been analyzed and understood well in the past, certain aspects of quantumMarkov chain...

Approximate counting via random optimization

4 ABSTRACT: Let F be a family of subsets of 1, . . . , n . We propose a simple randomized n algorithm to estimate the cardinality of F from the maximum weight of a subset XgF in n n 4 a random weighting of 1, . . . , n . The examples include enumeration of perfect matchings in graphs, bases in matroids, and Hamiltonian cycles in graphs. Q 1997 John Wiley & Sons, Inc. Ž . Random Struct. Alg., 11...

Universal recovery map for approximate Markov chains

A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the condi...

Approximate regenerative-block bootstrap for Markov chains