Abstract
We design a fully polynomial time approximation scheme (FPTAS) for counting the number of matchings (packings) in arbitrary 3-uniform hypergraphs of maximum degree three, referred to as (3,3)-hypergraphs. It is the first polynomial time approximation scheme for that problem, which includes also, as a special case, the 3D Matching counting problem for 3-partite (3,3)-hypergraphs. The proof technique of this paper uses the general correlation decay technique and a new combinatorial analysis of the underlying structures of the intersection graphs. The proof method could be also of independent interest.
Part of research of the 3rd and 4th authors done at Emory University, Atlanta and another part during their visits to the Institut Mittag-Leffler (Djursholm, Sweden).
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References
Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings. In: STOC 2007—Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 122–127. ACM (2007)
Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings (2008), http://people.math.gatech.edu/~tetali/PUBLIS/BGKNT_final.pdf
Beineke, L.W.: Characterizations of derived graphs. J. Combin. Theory 9, 129–135 (1970)
Chudnovsky, M., Seymour, P.: The roots of the independence polynomial of a clawfree graph. J. Combin. Theory Ser. B 97(3), 350–357 (2007)
Dobrushin, R.: Prescribing a system of random variables by conditional distributions. Theor. Probab. Appl. 15, 458–486 (1970)
Dyer, M., Frieze, A., Jerrum, M.: On counting independent sets in sparse graphs. SIAM J. Comput. 31(5), 1527–1541 (2002)
Fadnavis, S.: Approximating independence polynomials of claw-free graphs (2012), http://www.math.harvard.edu/~sukhada/IndependencePolynomial.pdf
Greenhill, C.: The complexity of counting colourings and independent sets in sparse graphs and hypergraphs. Comput. Complexity 9(1), 52–72 (2000)
Heilmann, O.: Existence of phase transitions in certain lattice gases with repulsive potential. Lett. Al Nuovo Cimento Series 2 3(3), 95–98 (1972)
Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18(6), 1149–1178 (1989)
Karpiński, M., Ruciński, A., Szymańska, E.: Approximate counting of matchings in sparse uniform hypergraphs. In: 2013 Proceedings of the Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 72–79. SIAM (2013)
Kelly, F.P.: Stochastic models of computer communication systems. J. Roy. Statist. Soc. Ser. B 47(3), 379–395, 415–428 (1985)
Luby, M., Vigoda, E.: Fast convergence of the Glauber dynamics for sampling independent sets. Random Structures Algorithms 15(3-4), 229–241 (1999)
Sly, A.: Computational transition at the uniqueness threshold. In: 2010 IEEE 51st Annual Symposium on Foundations of Computer Science FOCS 2010, pp. 287–296 (2010)
Sly, A., Sun, N.: The computational hardness of counting in two-spin models on d-regular graphs. In: FOCS, pp. 361–369 (2012), http://arxiv.org/abs/1203.2602
Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)
Weitz, D.: Counting independent sets up to the tree threshold. In: STOC 2006: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 140–149. ACM (2006)
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Dudek, A., Karpinski, M., Ruciński, A., Szymańska, E. (2014). Approximate Counting of Matchings in (3,3)-Hypergraphs. In: Ravi, R., Gørtz, I.L. (eds) Algorithm Theory – SWAT 2014. SWAT 2014. Lecture Notes in Computer Science, vol 8503. Springer, Cham. https://doi.org/10.1007/978-3-319-08404-6_33
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DOI: https://doi.org/10.1007/978-3-319-08404-6_33
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