Work-Efficient Parallel Counting via Sampling

Abstract: A canonical approach to approximating the partition function of a Gibbs distribution via sampling is simulated annealing. This method has led to efficient reductions from counting to sampling, including: $\bullet$ classic non-adaptive (parallel) algorithms with sub-optimal cost (Dyer-Frieze-Kannan '89; Bez\'akov\'a-\v{S}tefankovi\v{c}-Vazirani-Vigoda '08); $\bullet$ adaptive (sequential) algorithms with near-optimal cost (\v{S}tefankovi\v{c}-Vempala-Vigoda '09; Huber '15; Kolmogorov '18; Harris-Kolmogorov '24). We present an algorithm that achieves both near-optimal total work and efficient parallelism, providing a reduction from counting to sampling with logarithmic depth and near-optimal work. As consequences, we obtain work-efficient parallel counting algorithms for several important models, including the hardcore and Ising models within the uniqueness regime.