Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains

  title={Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains},
  author={Alistair Sinclair and Mark Jerrum},
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 polynomial time. It follows that, for self-reducible structures, polynomial time randomised algorithms for… 

Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow

  • A. Sinclair
  • Computer Science
    Combinatorics, Probability and Computing
  • 1992
A new upper bound on the mixing rate is presented, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph, and improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system.

Approximating the Permanent

A randomised approximation scheme for the permanent of a 0–1s presented, demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...

Disjoint Decomposition of Markov Chains and Sampling Circuits in Cayley Graphs

A new version of the decomposition theorem is presented where the pieces partition the state space, rather than forming a cover where pieces overlap, as was previously required, which is more natural and better suited to many applications.

Markov chains for sampling matchings

A lemma is developed which obtains better bounds on the mixing time in this case than existing theorems, in the case where β = 1 and the probability of a change in distance is proportional to the distance between the two states.

Relating counting complexity to non-uniform probability measures

A family of probability distributions over the set of solutions of a problem in TotP are presented, and it is proved that sampling and approximating the normalizing factor is easy; counting is equivalent to computing their normalizing factors.

Faster random generation of linear extensions

General mixing time bounds for finite Markov chains via the absolute spectral gap

We prove an upper bound on the total variation mixing time of a finite Markov chain in terms of the absolute spectral gap and the number of elements in the state space. Unlike results requiring

A Sequential Algorithm for Generating Random Graphs

A nearly-linear time algorithm for counting and randomly generating simple graphs with a given degree sequence in a certain range and it is shown that for d=O(n1/2−τ), the algorithm can generate an asymptotically uniform d-regular graph.

The Mixing of Markov Chains on Linear Extensions in Practice

The empirical results suggest that the Markov chain approach to sample linear extensions can be made to scale well in practice, provided that the actual mixing times can be realized by instance-sensitive bounds or termination rules.



Randomised algorithms for counting and generating combinatorial structures

The thesis studies the computational complexity of two natural classes of combinatorial problems: counting the elements of a finite set of structures and generating them uniformly at random. For each

Random Generation of Combinatorial Structures from a Uniform Distribution

The complexity of approximate counting

The complexity of computing approximate solutions to problems in #P is classified in terms of the polynomial-time hierarchy (for short, P-hierarchy) in order to study a class of restricted, but very natural, probabilistic sampling methods motivated by the particular counting problems.

Generating Random Unlabelled Graphs

  • N. Wormald
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 1987
This chapter discusses how to generate unlabelled graphs on n vertices uniformly at random, without calculating the total numbers of such graphs, but by using asymptotic enumeration results.

On the Markov Chain Simulation Method for Uniform Combinatorial Distributions and Simulated Annealing

  • D. Aldous
  • Mathematics
    Probability in the Engineering and Informational Sciences
  • 1987
Uniform distributions on complicated combinatorial sets can be simulated by the Markov chain method. A condition is given for the simulations to be accurate in polynomial time. Similar analysis of

The Uniform Selection of Free Trees

  • H. Wilf
  • Mathematics
    J. Algorithms
  • 1981

How hard is it to marry at random? (On the approximation of the permanent)

Al though finding a perfect matching is easy and finding a Hamil tonian circuit is hard, counting perfect matchings and counting Hamiltonian circuits is equally hard, as hard as computing the number of solutions of any problem in NP.

Difference equations, isoperimetric inequality and transience of certain random walks

The difference Laplacian on a square lattice in Rn has been stud- ied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many

Markov Chain Models--Rarity And Exponentiality

0. Introduction and Summary.- 1. Discrete Time Markov Chains Reversibility in Time.- 1.00. Introduction.- 1.0. Notation, Transition Laws.- 1.1. Irreducibility, Aperiodicity, Ergodicity Stationary