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We show that the rate of the correlation decay

Simple deterministic approximation algorithms for counting matchings

STOC, pp.122-127, (2007)

Cited: 108|Views32

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dl.acm.org dblp.uni-trier.de academic.microsoft.com

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deterministic algorithmtotal numberdegree graphtime expsimple deterministic approximation algorithmMore(11+)

Abstract

We construct a deterministic fully polynomial time approximationscheme (FPTAS) for computing the total number of matchings in abounded degree graph. Additionally, for an arbitrary graph, weconstruct a deterministic algorithm for computing approximately thenumber of matchings within running time exp(O(√n log2n)),where n is the number of ve...More

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Introduction

The focus of the paper is the problem of computing the total number of matchings of a given graph.
This problem, along with many other combinatorial counting problems falls into the class of #P complexity class, and modulo a basic complexity theoretic conjecture, cannot be solved in polynomial time.
Many randomized approximation schemes for various counting problems were derived in this way – see e.g., Vigoda’s survey [Vig00]

Highlights

The focus of the paper is the problem of computing the total number of matchings of a given graph. This problem, along with many other combinatorial counting problems falls into the class of #P complexity class, and modulo a basic complexity theoretic conjecture, cannot be solved in polynomial time
Many randomized approximation schemes for various counting problems were derived in this way – see e.g., Vigoda’s survey [Vig00]
The following questions is raised: is there a deterministic fully polynomial time approximation scheme for counting matchings for the class of all graphs? There are some fundamental limitations of the approach proposed in this paper the correlation decay rate corresponding to the case of matchings seems to be of order
It is of interest to see to what extent the correlation decay approach can be used for solving other type of counting problems, such as counting the number of spanning trees and forests, counting the number of feasible solutions of bin-packing problems, and a host of other problems where the Monte Carlo Markov chain method has been successful
This line of investigation might bring us a step closer to understanding the extent to which randomized algorithms are more powerful than deterministic algorithms

Conclusion

The authors have constructed a deterministic algorithm for counting approximately the number of matchings of a given graph. subexponential

The time aelxgpor(iOth(√mnrluongs2 in polynomial time for the class of bounded degree graphs, and n)) for the class of all graphs, where n is the number of nodes. in

Naturally, the following questions is raised: is there a deterministic FPTAS for counting matchings for the class of all graphs? There are some fundamental limitations of the approach proposed in this paper decay rate corresponding to the case of matchings seems to be of order

), and the authors speculate that the improvement should come along some combinatorial,, type arguments.
It is of interest to see to what extent the correlation decay approach can be used for solving other type of counting problems, such as counting the number of spanning trees and forests, counting the number of feasible solutions of bin-packing problems, and a host of other problems where the MCMC method has been successful
This line of investigation might bring them a step closer to understanding the extent to which randomized algorithms are more powerful than deterministic algorithms

Funding

Bayati, Nair and Tetali acknowledge the support of the Theory Group at Microsoft Research, where part of this work was carried out

Reference

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