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1. 35th Computational Complexity Conference (CCC 2020).

   Cai, Jin-Yi; Liu, Tianyu; Lu, Pinyan; Yu, Jing (July 2020, 35th Computational Complexity Conference (CCC 2020).)

   Saraf, Shubhangi (Ed.)

   We initiate a study of the classification of approximation complexity of the eight-vertex model defined over 4-regular graphs. The eight-vertex model, together with its special case the six-vertex model, is one of the most extensively studied models in statistical physics, and can be stated as a problem of counting weighted orientations in graph theory. Our result concerns the approximability of the partition function on all 4-regular graphs, classified according to the parameters of the model. Our complexity results conform to the phase transition phenomenon from physics. We introduce a quantum decomposition of the eight-vertex model and prove a set of closure properties in various regions of the parameter space. Furthermore, we show that there are extra closure properties on 4-regular planar graphs. These regions of the parameter space are concordant with the phase transition threshold. Using these closure properties, we derive polynomial time approximation algorithms via Markov chain Monte Carlo. We also show that the eight-vertex model is NP-hard to approximate on the other side of the phase transition threshold. 

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2. Counting Independent Sets in Unbalanced Bipartite Graphs

   https://doi.org/10.1137/1.9781611975994.88  

   Cannon, Sarah; Perkins, Will (January 2020, Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms)

   Chawla, Shuchi (Ed.)

   Understanding the complexity of approximately counting the number of weighted or unweighted independent sets in a bipartite graph (#BIS) is a central open problem in the field of approximate counting. Here we consider a subclass of this problem and 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 λ = 1 (approximating the number of independent sets of G) and Delta_R >= 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. Further consequences of this method for unbalanced bipartite graphs include an efficient sampling algorithm for the hard-core model and zero-freeness results for the partition function with complex fugacities. By utilizing connections between the cluster expansion and joint cumulants of certain random variables, we go beyond previous algorithmic applications of the cluster expansion to prove that the hard-core model exhibits exponential decay of correlations for all graphs and fugacities satisfying our conditions. This illustrates the applicability of statistical mechanics tools to algorithmic problems and refines our understanding of the connections between different methods of approximate counting. 

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3. Approximating Observables Is as Hard as Counting

   Galanis, Andreas; Štefankovič, Daniel; Vigoda, Eric (June 2022, ICALP 2022)

   We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. A recent work of Galanis et al (2021) established NP-hardness of approximating the average size of an independent set utilizing hardness of the corresponding optimization problem and the related phase transition behavior. We instead consider settings where the underlying optimization problem is easily solvable. Our main contribution is to classify the complexity of approximating a wide class of observables via a generic reduction from approximate counting to the problem of estimating local observables. The key idea is to use the observables to interpolate the counting problem. Using this new approach, we are able to study observables on bipartite graphs where the underlying optimization problem is easy but the counting problem is believed to be hard. The most-well studied class of graphs that was excluded from previous hardness results were bipartite graphs. We establish hardness for estimating the average size of the independent set in bipartite graphs of maximum degree 6; more generally, we show tight hardness results for general vertex-edge observables for antiferromagnetic 2-spin systems on bipartite graphs. Our techniques go beyond 2-spin systems, and for the ferromagnetic Potts model we establish hardness of approximating the number of monochromatic edges in the same region as known hardness of approximate counting results. 

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4. Quadratic Speedups in Parallel Sampling from Determinantal Distributions

   https://doi.org/10.1145/3558481.3591104  

   Anari, Nima; Burgess, Callum; Tian, Kevin; Vuong, Thuy-Duong (June 2023, ACM)

   We study the problem of parallelizing sampling from distributions related to determinants: symmetric, nonsymmetric, and partition-constrained determinantal point processes, as well as planar perfect matchings. For these distributions, the partition function, a.k.a.\ the count, can be obtained via matrix determinants, a highly parallelizable computation; Csanky proved it is in NC. However, parallel counting does not automatically translate to parallel sampling, as classic reductions between the two are inherently sequential. We show that a nearly quadratic parallel speedup over sequential sampling can be achieved for all the aforementioned distributions. If the distribution is supported on subsets of size $$k$$ of a ground set, we show how to approximately produce a sample in $$\widetilde{O}(k^{\frac{1}{2} + c})$$ time with polynomially many processors for any constant $c>0$. In the two special cases of symmetric determinantal point processes and planar perfect matchings, our bound improves to $$\widetilde{O}(\sqrt k)$$ and we show how to sample exactly in these cases. As our main technical contribution, we fully characterize the limits of batching for the steps of sampling-to-counting reductions. We observe that only $O(1)$ steps can be batched together if we strive for exact sampling, even in the case of nonsymmetric determinantal point processes. However, we show that for approximate sampling, $$\widetilde{\Omega}(k^{\frac{1}{2}-c})$$ steps can be batched together, for any entropically independent distribution, which includes all mentioned classes of determinantal point processes. Entropic independence and related notions have been the source of breakthroughs in Markov chain analysis in recent years, so we expect our framework to prove useful for distributions beyond those studied in this work. 

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5. A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights

   Govorov, Artem Cai (July 2020, 47th International Colloquium on Automata, Languages, and Programming)

   Czumaj, Artur Dawar (Ed.)

   We consider the complexity of counting weighted graph homomorphisms defined by a symmetric matrix A. Each symmetric matrix A defines a graph homomorphism function Z A (·), also known as the partition function. Dyer and Greenhill [10] established a complexity dichotomy of Z A (·) for symmetric {0, 1}-matrices A, and they further proved that its #P-hardness part also holds for bounded degree graphs. Bulatov and Grohe [4] extended the Dyer-Greenhill dichotomy to nonnegative symmetric matrices A. However, their hardness proof requires graphs of arbitrarily large degree, and whether the bounded degree part of the Dyer-Greenhill dichotomy can be extended has been an open problem for 15 years. We resolve this open problem and prove that for nonnegative symmetric A, either Z A (G) is in polynomial time for all graphs G, or it is #P-hard for bounded degree (and simple) graphs G. We further extend the complexity dichotomy to include nonnegative vertex weights. Additionally, we prove that the #P-hardness part of the dichotomy by Goldberg et al. [12] for Z A (·) also holds for simple graphs, where A is any real symmetric matrix. 

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