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# Simple deterministic approximation algorithms for counting matchings

STOC, pp.122-127, (2007)

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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

- A. Bandyopadhyay and D. Gamarnik, Counting without sampling: New algorithms for enumeration problems using statistical physics, Proceedings of 17th ACM-SIAM Symposium on Discrete Algorithms (SODA) (2006).
- [BKMP01] N. Berger, C. Kenyon, E. Mossel, and Y. Peres, Glauber dynamics on trees and hyperbolic graphs, Proc. 42nd IEEE Symposium on Foundations of Computer Science (FOCS) (2001).
- M. Bayati and C. Nair, A rigorous proof of the cavity method for counting matchings, Annual Allerton Conference on Communication, Control and Computing, 2006.
- [Dob70] R. L. Dobrushin, Prescribing a system of random variables by the help of conditional distributions, Theory of Probability and its Applications 15 (1970), 469–497.
- [DSVW02] M. Dyer, A. Sinclair, E. Vigoda, and D. Weitz, Mixing in time and space for lattice spin systems: a combinatorial view, Random Struct. & Alg. 24 (2004), pp.461–479.
- D. Gamarnik and D. Katz, Correlation decay and deterministic FPTAS for counting list-colorings of a graph, Proceedings of 18th ACM-SIAM Symposium on Discrete Algorithms (SODA) 2007.
- [GMP05] L. A. Goldberg, R. Martin, and M. Paterson, Strong spatial mixing with fewer colors for lattice graphs, SIAM J. Comput. 35 (2005), no. 2, 486–517.
- [God81] C. D. Godsil, Matchings and walks in graphs, J. Graph Th. 5 (1981), 285–297.
- M. Jerrum, Counting, sampling and integrating: algorithms and complexity. Lecture notes, Chapter 3., 2006.
- M. Jerrum and A. Sinclair, The Markov chain Monte Carlo method: an approach to approximate counting and integration, Approximation algorithms for NP-hard problems (D. Hochbaum, ed.), PWS Publishing Company, Boston, MA, 1997.
- M. Jerrum, A. Sinclair, and E. Vigoda, A polynomial-time approximation algorithms for permanent of a matrix with non-negative entries, Journal of the Association for Computing Machinery 51 (2004), no. 4, 671–697.
- [JVV86] M. Jerrum, L. Valiant, and V. Vazirani, Random generation of combinatorial structures from a uniform distribution, Theoret. Comput. Sci. 43 (1986), no. 2-3, 169188.
- [LSW00] N. Linial, A. Samorodnitsky, and A. Wigderson, A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents, Combinatorica 20 (2000), no. 4, 545–568.
- A. Montanari and G. Semerjian, Rigorous inequalities between length and time scales in glassy systems, Preprint in arXiv.org (2006).
- S. Tatikonda and M.I. Jordan, Loopy belief propagation and Gibbs measures, In Uncertainty in Artificial Intelligence (UAI), D. Koller and A. Darwiche (Eds)., 2002.
- L. G. Valiant, The complexity of computing the permanent, Theoretical computer science 8 (1979), 189–201.
- [Vig00] E. Vigoda, Improved bounds for sampling colorings, Journal of Mathematical Physics (2000).
- [Wei06] D. Weitz, Counting down the tree, Proc. 38th Ann. Symposium on the Theory of Computing (STOC) (2006).

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