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1. 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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2. 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).

   Cai, Jin-Yi Liu (July 2020, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).)

   Czumaj, Artur (Ed.)

   We study the approximation complexity of the partition function of the eight-vertex model on general 4-regular graphs. For the first time, we relate the approximability of the eight-vertex model to the complexity of approximately counting perfect matchings, a central open problem in this field. Our results extend those in [8]. In a region of the parameter space where no previous approximation complexity was known, we show that approximating the partition function is at least as hard as approximately counting perfect matchings via approximation-preserving reductions. In another region of the parameter space which is larger than the region that is previously known to admit Fully Polynomial Randomized Approximation Scheme (FPRAS), we show that computing the partition function can be reduced to counting perfect matchings (which is valid for both exact and approximate counting). Moreover, we give a complete characterization of nonnegatively weighted (not necessarily planar) 4-ary matchgates, which has been open for several years. The key ingredient of our proof is a geometric lemma. We also identify a region of the parameter space where approximating the partition function on planar 4-regular graphs is feasible but on general 4-regular graphs is equivalent to approximately counting perfect matchings. To our best knowledge, these are the first problems that exhibit this dichotomic behavior between the planar and the nonplanar settings in approximate counting. 

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3. A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs

   https://doi.org/10.4230/LIPIcs.ICALP.2024.35  

   Carlson, Charlie; Davies, Ewan; Kolla, Alexandra; Potukuchi, Aditya (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph - in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to "high-temperature" problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. Our contributions include a novel extension of the method of graph containers that differs considerably from other recent low-temperature algorithms. The additional key insights come from spectral graph theory and have previously been successful in approximation algorithms. As a result, we can overcome some limitations that seem inherent to the aforementioned class of algorithms. In particular, we exploit the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e., bounded threshold rank), which, via subspace enumeration, lets us enumerate small cuts efficiently. 

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4. Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers

   https://doi.org/10.1145/3593809  

   Jambulapati, Arun; Sidford, Aaron (February 2024, ACM Transactions on Algorithms)

   In this paper we provide anO(mloglog^O(1)nlog (1/ϵ))-expected time algorithm for solving Laplacian systems onn-nodem-edge graphs, improving upon the previous best expected runtime of\(O(m \sqrt {\log n} \mathrm{log log}^{O(1)} n \log (1/\epsilon)) \)achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of low spectral stretch graph approximations with improved stretch and sparsity bounds. As motivation for this work, we show that for every set of vectors in\(\mathbb {R}^d \)(not just those induced by graphs) and all integerk> 1 there exist an ultra-sparsifier withd− 1 +O(d/k) re-weighted vectors of relative condition number at mostk². For smallk, this improves upon the previous best known multiplicative factor of\(k \cdot \tilde{O}(\log d) \), which is only known for the graph case. Additionally, in the graph case we employ our low-stretch subgraph construction to obtainn− 1 +O(n/k)-edge ultrasparsifiers of relative condition numberk^{1 +o(1)}fork=ω(log ^δn) for anyδ> 0: this improves upon the previous work fork=o(exp (log ^{1/2 −δ}n)). 

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5. Sampling List Packings

   https://doi.org/10.4230/LIPIcs.ITCS.2025.24  

   Camrud, Evan; Davies, Ewan; Karduna, Alex; Lee, Holden (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   We initiate the study of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted substantial attention. For list packing the setup is similar, but we seek a full decomposition of the lists of colors into pairwise-disjoint proper list colorings. The existence of a list packing implies the existence of a list coloring, but the converse is false. Recent works on list packing have focused on existence or extremal results of on the number of list packings, but here we turn to the algorithmic aspects of counting and sampling. In graphs of maximum degree Δ and when the number of colors is at least Ω(Δ²), we give a fully polynomial-time randomized approximation scheme (FPRAS) based on rapid mixing of a natural Markov chain (the Glauber dynamics) which we analyze with the path coupling technique. Some motivation for our work is the investigation of an atypical spin system, one where the number of spins for each vertex is much larger than the graph degree. 

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