Counting independent sets in unbalanced bipartite graphs

We give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides $(L,R)$ of the bipartition. This includes, among others, the biregular case when $\lambda=1$ (approximating the number of independent sets of $G$) and $\Delta_R \geq 7\Delta_L \log(\Delta_L)$. Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition. As a consequence of the method, we also prove that the hard-core model on such graphs exhibits exponential decay of correlations by utilizing connections between the cluster expansion and joint cumulants.

[1]  David Preiss,et al.  Cluster expansion for abstract polymer models , 1986 .

[2]  R. L. Dobrushin,et al.  Estimates of semiinvariants for the Ising model at low temperatures , 1996 .

[3]  Albert N. Shiryaev,et al.  On a Method of Calculation of Semi-Invariants , 1959 .

[4]  László Lovász,et al.  Left and right convergence of graphs with bounded degree , 2010, Random Struct. Algorithms.

[5]  Will Perkins,et al.  Algorithmic Pirogov–Sinai theory , 2019, Probability Theory and Related Fields.

[6]  Leslie Ann Goldberg,et al.  Approximating the partition function of the ferromagnetic Potts model , 2010, JACM.

[7]  Dror Weitz,et al.  Counting independent sets up to the tree threshold , 2006, STOC '06.

[8]  Will Perkins,et al.  Algorithms for #BIS-hard problems on expander graphs , 2018, SODA.

[9]  Alexander I. Barvinok,et al.  Weighted counting of solutions to sparse systems of equations , 2017, Combinatorics, Probability and Computing.

[10]  Allan Sly,et al.  Computational Transition at the Uniqueness Threshold , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[11]  Andreas Björklund,et al.  Computing the Tutte Polynomial in Vertex-Exponential Time , 2007, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[12]  Eric Vigoda,et al.  Danielštefankovič ‡ , 2022 .

[13]  Alexander I. Barvinok,et al.  Combinatorics and Complexity of Partition Functions , 2017, Algorithms and combinatorics.

[14]  Jiabao Lin,et al.  Counting independent sets and colorings on random regular bipartite graphs , 2019, APPROX-RANDOM.

[15]  R. Dobrushin,et al.  Completely analytical interactions: Constructive description , 1987 .

[16]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

[17]  Allan Sly,et al.  Counting in two-spin models on d-regular graphs , 2014 .

[18]  Yvan Alain Velenik,et al.  Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction , 2017 .

[19]  H. Kunz,et al.  General properties of polymer systems , 1971 .

[20]  V. Malyshev.  CLUSTER EXPANSIONS IN LATTICE MODELS OF STATISTICAL PHYSICS AND THE QUANTUM THEORY OF FIELDS , 1980 .

[21]  R. L. Dobrushin,et al.  Perturbation methods of the theory of Gibbsian fields , 1996 .

[22]  Eric Vigoda,et al.  #BIS-Hardness for 2-Spin Systems on Bipartite Bounded Degree Graphs in the Tree Non-uniqueness Region , 2014, APPROX-RANDOM.

[23]  Martin E. Dyer,et al.  The Relative Complexity of Approximate Counting Problems , 2000, Algorithmica.

[24]  Eric Vigoda,et al.  Improved inapproximability results for counting independent sets in the hard-core model , 2014, Random Struct. Algorithms.