FPTAS for Counting Weighted Edge Covers
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 . 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  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 . Our algorithm also goes beyond that to certain generalization of edge cover.
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- 1.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)Google Scholar
- 3.Cai, J.-Y., Lu, P., Xia, M.: Holant problems and counting CSP. In: Proceedings of STOC, pp. 715–724 (2009)Google Scholar
- 6.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)CrossRefGoogle Scholar
- 7.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)Google Scholar
- 8.Li, L., Lu, P., Yin, Y.: Approximate counting via correlation decay in spin systems. In: Proceedings of SODA, pp. 922–940. SIAM (2012)Google Scholar
- 9.Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. In: Proceedings of SODA, pp. 67–84 (2013)Google Scholar
- 10.Lin, C., Liu, J., Lu, P.: A simple FPTAS for counting edge covers. In: Proceedings of SODA, pp. 341–348 (2014)Google Scholar
- 11.Liu, J., Lu, P.: FPTAS for counting monotone CNF. arXiv preprint arXiv:1311.3728 (2013)Google Scholar
- 14.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)Google Scholar
- 15.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)Google Scholar
- 16.Sly, A.: Computational transition at the uniqueness threshold. In: Proceedings of FOCS, pp. 287–296. IEEE (2010)Google Scholar
- 17.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)Google Scholar
- 20.Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of STOC, pp. 140–149. ACM (2006)Google Scholar