Abstract
An edge cover of a graph is a set of edges in which each vertex has at least one of its incident edges. The problem of counting the number of edge covers is #P-complete and was shown to admit a fully polynomial-time approximation scheme (FPTAS) recently [10]. Counting weighted edge covers is the problem of computing the sum of the weights for all the edge covers, where the weight of each edge cover is defined to be the product of the edge weights of all the edges in the cover. The FPTAS in [10] cannot apply to general weighted counting for edge covers, which was stated as an open question there. Such weighted counting is generally interesting as for instance the weighted counting independent sets (vertex covers) problem has been exhaustively studied in both statistical physics and computer science. Weighted counting for edge cover is especially interesting as it is closely related to counting perfect matchings, which is a long-standing open question. In this paper, we obtain an FPTAS for counting general weighted edge covers, and thus solve an open question in [10]. Our algorithm also goes beyond that to certain generalization of edge cover.
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References
Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings. In: Proceedings of STOC, pp. 122–127. ACM (2007)
Bezáková, I., Rummler, W.A.: Sampling edge covers in 3-regular graphs. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 137–148. Springer, Heidelberg (2009)
Cai, J.-Y., Lu, P., Xia, M.: Holant problems and counting CSP. In: Proceedings of STOC, pp. 715–724 (2009)
Cai, J.-Y., Lu, P., Xia, M.: Computational complexity of Holant problems. SIAM Journal on Computing 40(4), 1101–1132 (2011)
Dyer, M., Frieze, A., Jerrum, M.: On counting independent sets in sparse graphs. SIAM Journal on Computing 31(5), 1527–1541 (2002)
Galanis, A., Ge, Q., Štefankovič, D., Vigoda, E., Yang, L.: Improved inapproximability results for counting independent sets in the hard-core model. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) RANDOM 2011 and APPROX 2011. LNCS, vol. 6845, pp. 567–578. Springer, Heidelberg (2011)
Jerrum, M., Sinclair, A.: The Markov chain Monte Carlo method: an approach to approximate counting and integration. In: Approximation Algorithms for NP-hard Problems, pp. 482–520 (1996)
Li, L., Lu, P., Yin, Y.: Approximate counting via correlation decay in spin systems. In: Proceedings of SODA, pp. 922–940. SIAM (2012)
Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. In: Proceedings of SODA, pp. 67–84 (2013)
Lin, C., Liu, J., Lu, P.: A simple FPTAS for counting edge covers. In: Proceedings of SODA, pp. 341–348 (2014)
Liu, J., Lu, P.: FPTAS for counting monotone CNF. arXiv preprint arXiv:1311.3728 (2013)
Lu, P., Wang, M., Zhang, C.: FPTAS for weighted Fibonacci gates and its applications. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 787–799. Springer, Heidelberg (2014)
Restrepo, R., Shin, J., Tetali, P., Vigoda, E., Yang, L.: Improved mixing condition on the grid for counting and sampling independent sets. Probability Theory and Related Fields 156(1-2), 75–99 (2013)
Sinclair, A., Srivastava, P., Thurley, M.: Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. In: Proceedings of SODA, pp. 941–953. SIAM (2012)
Sinclair, A., Srivastava, P., Yin, Y.: Spatial mixing and approximation algorithms for graphs with bounded connective constant. In: Proceedings of FOCS, pp. 300–309. IEEE (2013)
Sly, A.: Computational transition at the uniqueness threshold. In: Proceedings of FOCS, pp. 287–296. IEEE (2010)
Sly, A., Sun, N.: The computational hardness of counting in two-spin models on d-regular graphs. In: Proceedings of FOCS, pp. 361–369. IEEE (2012)
Valiant, L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM Journal on Computing 31(4), 1229–1254 (2002)
Valiant, L.G.: Holographic algorithms. SIAM Journal on Computing 37(5), 1565–1594 (2008)
Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of STOC, pp. 140–149. ACM (2006)
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Liu, J., Lu, P., Zhang, C. (2014). FPTAS for Counting Weighted Edge Covers. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_54
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DOI: https://doi.org/10.1007/978-3-662-44777-2_54
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