Approximating the Permanent

@article{Jerrum1989ApproximatingTP,
  title={Approximating the Permanent},
  author={Mark Jerrum and Alistair Sinclair},
  journal={SIAM J. Comput.},
  year={1989},
  volume={18},
  pages={1149-1178}
}
A randomised approximation scheme for the permanent of a 0–1s presented. The task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph; the latter is accomplished by simulating a Markov chain whose states are the matchings in the graph. For a wide class of 0–1 matrices the approximation scheme is fully-polynomial, i.e., runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output. This class… 

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References

SHOWING 1-10 OF 29 REFERENCES

Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains

TLDR
The general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods: this in turn leads to randomised approximate counting algorithms for these graphs with very good asymptotic behaviour.

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

TLDR
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.

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

Polytopes, permanents and graphs with large factors

TLDR
It is proved that for any graph G the k-slice of the well-known Edmonds matching polytope has magnification 1, and it is proven that the ratio of the number of almost perfect matchings to the number-of-perfect matchings is at most n/sup 3n/d.

Convergence of an annealing algorithm

TLDR
This paper presents a model of the annealing algorithm and proves that the algorithm converges with probability arbitrarily close to 1, and shows that there are cases where convergence takes exponentially long—that is, it is no better than a deterministic method.

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

The time complexity of maximum matching by simulated annealing

TLDR
It is shown for arbitrary graphs that a degenerate form of the basic annealing algorithm (obtained by letting “temperature” be a suitably chosen constant) produces matchings with nearly maximum cardinality in polynomial average time.

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

Bounds on the ² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality

We prove a general version of Cheeger's inequality for discrete-time Markov chains and continuous-time Markovian jump processes, both reversible and nonreversible, with general state space. We also

Monte-Carlo algorithms for enumeration and reliability problems

  • R. KarpM. Luby
  • Mathematics
    24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
  • 1983
TLDR
A simple but very general Monte-Carlo technique for the approximate solution of enumeration and reliability problems and several applications are given.