approximate partition function

Approximate computation of partition functions for spin systems has traditionally been limited by quadratic time complexity. This work establishes, for the first time, a profound connection between subquadratic-time approximate counting and perfect marginal sampling, introducing a black-box reduction framework that integrates low-variance marginal estimation with sublinear sampling techniques. The framework accelerates approximate counting for a broad class of spin systems to Õ(n²⁻ᵟ) time for some constant δ > 0. This advance substantially expands the parameter regimes amenable to efficient approximation: for instance, in the hard-core model, it raises the fugacity threshold from o(Δ⁻³/²) to 1/(Δ−1), and achieves breakthrough speedups for the Ising model, hypergraph independent sets, and vertex coloring problems.

This work presents the first deterministic fully polynomial-time approximation scheme (FPTAS) for spin systems satisfying coupling independence and bounded-degree (maximum degree Δ) constraints, breaking reliance on randomized algorithms. Methodologically, it introduces a novel deterministic counting framework built upon recursive decomposition, tree-like structure approximation, and rigorous coupling independence analysis, with tight control over error propagation. Key contributions include the first three deterministic FPTASes for the q-coloring problem: (1) for graphs with Δ ≥ 3 and q ≥ (11/6 − ε₀)Δ; (2) for Δ ≥ 125 and q ≥ 1.809Δ; and (3) for graphs of large girth and q ≥ Δ + 3. All degree–coloring thresholds match the best known bounds achieved by randomized algorithms, marking a fundamental advance in deterministic approximate counting for graph coloring.

This paper presents the first efficient distributed approximation algorithm for computing the partition function of graphical Gibbs distributions in the Congested Clique model, unifying counting problems for combinatorial structures such as $q$-colorings and independent sets. Methodologically, it introduces, for the first time in this model, the theoretical connection between combinatorial sampling and counting, establishing a generic framework grounded in locality and rapid mixing conditions; it further enables parallel Markov chain sampling via triangle counting and semiring matrix multiplication. Theoretical contributions include: (i) an $widetilde{O}(n^{1/3}/varepsilon^2)$-round algorithm for $q$-coloring counting when $q > 2Delta$; and (ii) convergence to $varepsilon$-accuracy for the hard-core model within $widetilde{O}(1/varepsilon^2)$ rounds under fugacity $lambda leq alpha/(Delta-1)$ with $alpha < 1$, substantially improving upon prior distributed results.

This paper addresses the approximate counting of partition functions for spin systems. We propose two randomized approximation algorithms achieving high-precision estimation in subquadratic time. Methodologically, we overcome Weitz’s algorithm’s stringent reliance on correlation decay by integrating random sampling, aggregation-based approximation, and spatial mixing analysis—tailoring time bounds to graph growth properties (e.g., quadratic growth on ℤ², polynomial growth on ℤᵈ). For sparse hard-core models and planar graphs under strong spatial mixing (SSM), we achieve ε-approximation in Õ(n²⁻ᶜ/ε²) time—the first such subquadratic guarantee within the SSM threshold. On ℤᵈ lattices, our bound improves to Õ(n²ε⁻²/2ᶜ(log n)¹⁄ᵈ), substantially outperforming standard O(n²) approaches. Our key contributions are: (i) the first subquadratic partition function estimation algorithm for polynomially growing graph families; and (ii) an extended applicability regime up to activity λ = O(Δ⁻¹·⁵⁻ᶜ¹), surpassing prior limits.

This paper addresses the problem of efficiently approximating the partition function of Gibbs distributions in parallel. Existing algorithms face a fundamental trade-off between total work and parallel depth. We propose the first counting-to-sampling reduction framework that simultaneously achieves near-optimal total work—approaching the information-theoretic lower bound—and logarithmic parallel depth. Methodologically, we build upon the simulated annealing paradigm, designing an adaptive temperature schedule and a parallel sampling verification mechanism to ensure work efficiency. Our theoretical contribution is the first work-efficient parallel counting algorithm for both the hardcore model and the Ising model within their uniqueness regimes. This overcomes the inherent limitations of prior approaches: non-adaptive algorithms suffer from suboptimal work complexity, while adaptive ones are inherently sequential. Our framework thus unifies asymptotic optimality in work with high parallelism, establishing a new state of the art in parallel approximate counting.

This work investigates the complexity of approximately counting configurations in two-spin systems—such as the hard-core model and graph coloring—on planar graphs. By integrating computational complexity theory, probabilistic approximation algorithms, and AI-assisted mathematical reasoning (GPT-5.6 Sol Ultra), it establishes for the first time a complete characterization of the necessary and sufficient conditions under which a fully polynomial randomized approximation scheme (FPRAS) exists in the regime of small external fields or small activity parameters. The main contributions include proving the existence of an FPRAS for the hard-core model with small activity parameters and demonstrating that approximate counting of proper $q$-colorings on planar graphs is NP-hard for $q \geq 4$, thereby providing a full criterion for the existence of FPRAS in this class of problems.

This study addresses the lack of efficient sampling and partition function approximation algorithms for ferromagnetic two-state systems within certain parameter regimes. By constructing weighted subgraph and random-cluster-like models, the authors establish novel equivalences between these combinatorial structures and the target physical system. Leveraging these connections, they propose the first efficient sampling algorithm tailored to this regime and design a partition function approximation algorithm running in nearly quadratic time on bounded-degree graphs and in polynomial time on general graphs. This represents a significant improvement over the method of Guo et al. (2020). The work integrates techniques from graph theory, statistical physics, and randomized approximation to deliver an enhanced computational framework for analyzing such systems.

This work addresses #P-hard counting problems—such as counting independent sets in general graphs and #2-SAT—that are inapproximable in polynomial time and prohibitively expensive to solve exactly. The authors propose a novel framework based on bounded, unweighted self-reducibility, which recursively decomposes problem instances and aggregates upper bounds from subproblems at a square-root recursion depth. By integrating enumeration with a hybrid sampling estimator, the approach substantially reduces the base of the exponential time complexity. The method achieves improved runtimes of O*(1.1869ⁿ) for independent set counting and O*(1.2373ⁿ) for #2-SAT approximation, outperforming the best known exact algorithms. It further extends to counting maximum cliques, minimal separators, and perfect matchings in subcubic graphs, and admits black-box quantum speedup.