Abstract
We prove rapid mixing of well-known Markov chains for the hardcore model on a new graph class, the class of chordal graphs with a bound on minimal separator size. In the hardcore model, for a given graph G and a fugacity parameter \(\lambda \in \mathbb {R}^+\), the goal is to produce an independent set S of G with probability proportional to \(\lambda ^{|S|}\). In general graphs and arbitrary \(\lambda \), producing a sample from this distribution in polynomial time is provably difficult. However, natural Markov chains converge to the correct distribution for any graph, leading to the study of their mixing times for different graph classes. Rapid mixing for graphs of bounded degrees and a range of \(\lambda \)s dependent on the maximum degree has attracted attention since the 1990s. Recent results showed rapid mixing for arbitrary \(\lambda \) and two other classes of graphs: graphs of bounded treewidth and graphs of bounded bipartite pathwidth. In this work, we extend these results by showing rapid mixing in a new graph class, class of chordal graphs with bounded minimal separators. Graphs in this class have no bound on the vertex degrees, the treewidth, or the bipartite pathwidth. Similar to the results dealing with bounded treewidth and with bounded bipartite pathwidth, we prove rapid mixing using the canonical paths technique. However, unlike in the previous works, we need to process the data using a non-linear, tree-like, approach.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This can be seen by taking a complete binary tree of \(\sqrt{n}\) vertices, where each vertex is replaced by a clique of size \(\sqrt{n}\), and each pair of adjacent cliques is connected by an edge.
References
Anari, N., Liu, K., Gharan, S.O.: Spectral independence in high-dimensional expanders and applications to the hardcore model. In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC). ACM (2020)
Bordewich, M., Kang, R.J.: Subset Glauber dynamics on graphs, hypergraphs and matroids of bounded tree-width. Electr. J. Comb. 21(4), P4.19 (2014)
Buneman, P., et al.: A characterisation of rigid circuit graphs. Discrete Math. 9(3), 205–212 (1974)
Dyer, M., Greenhill, C.: On Markov chains for independent sets. J. Algorithms 35(1), 17–49 (2000)
Dyer, M., Greenhill, C., Müller, H.: Counting independent sets in graphs with bounded bipartite pathwidth. In: Sau, I., Thilikos, D.M. (eds.) WG 2019. LNCS, vol. 11789, pp. 298–310. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-30786-8_23
Dyer, M.E., Frieze, A.M., Jerrum, M.: On counting independent sets in sparse graphs. SIAM J. Comput. 31(5), 1527–1541 (2002)
Efthymiou, C., Hayes, T.P., Štefankovič, D., Vigoda, E., Yin, Y.: Convergence of MCMC and loopy BP in the tree uniqueness region for the hard-core model. SIAM J. Comput. 48(2), 581–643 (2019)
Galanis, A., Ge, Q., Štefankovič, D., Vigoda, E., Yang, L.: Improved inapproximability results for counting independent sets in the hard-core model. Random Struct. Algorithms 45(1), 78–110 (2014)
Galanis, A., Štefankovič, D., Vigoda, E.: Inapproximability of the partition function for the antiferromagnetic ising and hard-core models. Comb. Probab. Comput. 25(4), 500–559 (2016)
Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory Ser. B 16(1), 47–56 (1974)
Ge, Q., Štefankovič, D.: A graph polynomial for independent sets of bipartite graphs. Comb. Probab. Comput. 21(5), 695–714 (2012)
Jerrum, M.: Counting, Sampling and Integrating: Algorithms and Complexity. Birhäuser (2003)
Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18(6), 1149–1178 (1989)
Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT press, Cambridge (2009)
Luby, M., Vigoda, E.: Fast convergence of the Glauber dynamics for sampling independent sets. Random Struct. Algorithms 15(3–4), 229–241 (1999)
Matthews, J.: Markov chains for sampling matchings. Ph.D. Thesis, University of Edinburgh (2008)
Mossel, E., Weitz, D., Wormald, N.: On the hardness of sampling independent sets beyond the tree threshold. Probab. Theory Rel. Fields 143, 401–439 (2009)
Okamoto, Y., Uno, T., Uehara, R.: Linear-time counting algorithms for independent sets in chordal Graphs. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 433–444. Springer, Heidelberg (2005). https://doi.org/10.1007/11604686_38
Sinclair, A.: Improved bounds for mixing rates of Markov chains and multicommodity flow. Comb. Prob. Comput. 1(4), 351–370 (1992)
Sly, A.: Computational transition at the uniqueness threshold. In: 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pp. 287–296. IEEE (2010)
Sly, A., Sun, N.: Counting in two-spin models on \(d\)-regular graphs. Ann. Probab. 42(6), 2383–2416 (2014)
Vandenberghe, L., Andersen, M.S., et al.: Chordal graphs and semidefinite optimization. Found. Trends ® Optim. 1(4), 241–433 (2015)
Vigoda, E.: A note on the Glauber dynamics for sampling independent sets. Electr. J. Comb. 8(1), R8 (2001)
Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of the 38th Annual ACM symposium on Theory of Computing (STOC), pp. 140–149 (2006)
Acknowledgements
We thank the anonymous referees for their valuable feedback. Research supported in part by NSF grant DUE-1819546.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Bezáková, I., Sun, W. (2020). Mixing of Markov Chains for Independent Sets on Chordal Graphs with Bounded Separators. In: Kim, D., Uma, R., Cai, Z., Lee, D. (eds) Computing and Combinatorics. COCOON 2020. Lecture Notes in Computer Science(), vol 12273. Springer, Cham. https://doi.org/10.1007/978-3-030-58150-3_54
Download citation
DOI: https://doi.org/10.1007/978-3-030-58150-3_54
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-58149-7
Online ISBN: 978-3-030-58150-3
eBook Packages: Computer ScienceComputer Science (R0)