deterministic approximate counting

This work studies efficient approximation of two natural counting problems under the Lovász Local Lemma (LLL) framework: the probability of the intersection of bad events and the dimension of the intersection of subspaces. Specifically, it addresses counting satisfying assignments for classical CNF formulas and counting the dimension of the satisfying subspace in quantum SAT. We propose a unified approximation framework based on cluster expansion. Our contributions include: (i) the first fully polynomial-time approximation scheme (FPTAS) for commuting projection operators; (ii) for general (non-commuting) projections, FPTAS under either inclusion–exclusion stability or spectral gap conditions, along with a novel affine approximation paradigm. The approach integrates cluster expansion, inclusion–exclusion principles, spectral analysis, and quantum satisfiability modeling—breaking reliance on commutativity or stringent constraint assumptions. This significantly extends the applicability of the LLL to counting problems beyond traditional limitations.

This work addresses the efficient (1+ε)-approximation of the number of occurrences of permutation patterns of length k ≤ 5 in real-valued sequences. While exact counting is computationally prohibitive, we present the first deterministic near-linear-time algorithm with time complexity O(n log n / ε²). Methodologically, we introduce Birgé’s distribution decomposition—previously unexplored in permutation pattern counting—integrated with a divide-and-conquer framework and discrete geometric embedding. This synergy enables the first provable separation between approximate and exact counting complexities. Our approach breaks known lower-bound barriers for k ≤ 5 and, empirically, achieves significantly faster runtime than exact algorithms for k = 4. Beyond improving asymptotic efficiency, this work pioneers the application of distribution testing techniques to combinatorial pattern counting, opening a new methodological avenue at the intersection of property testing, computational geometry, and enumerative combinatorics.

This paper addresses the model counting problem #nFBDD for nondeterministic read-once branching programs (nFBDDs), which is #P-hard and previously admitted only quasi-polynomial-time randomized approximation algorithms. We present the first fully polynomial randomized approximation scheme (FPRAS) for #nFBDD, breaking a long-standing complexity barrier. Our key innovation is a novel sampling dependency analysis technique that leverages the structural properties of nFBDDs to construct an efficiently mixable, tunable Markov chain. This enables controllable-error estimation of the number of satisfying assignments. The algorithm runs in time polynomial in both the input size and ε⁻¹, where ε is the desired additive error bound, and provides rigorous theoretical guarantees on approximation accuracy and convergence. Our approach significantly improves both the asymptotic efficiency and practical applicability of approximate model counting for nFBDDs.

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 work addresses automatic runtime and variable-size bound analysis for integer programs, focusing on the decidable subclass of periodic rational-solvable loops (PRS-loops). The proposed method introduces a modular analysis framework: it first derives local bounds for PRS-loops, then lifts them to global bounds via program transformation and inductive reasoning. Crucially, it extends the decidability of PRS-loop analysis to arbitrary integer programs by designing a synergistic synthesis mechanism combining abstract interpretation with rational linear algebra. The approach is fully automated in the tool KoAT, supporting precise derivation of polynomial and exponential complexity bounds as well as variable growth bounds. Experimental evaluation demonstrates effectiveness on diverse nontrivial integer programs, significantly improving the completeness, precision, and practicality of automated complexity analysis.

This work addresses the limitation in hardness proofs for path-packing problems that rely on randomized weight assignments by introducing the first deterministic variant of the isolation lemma. Combining combinatorial constructions with algebraic techniques, the authors explicitly design deterministic weights and employ formal verification to guarantee their correctness. This approach successfully eliminates probabilistic assumptions from several known hardness results, replacing randomized assignments with fully deterministic ones. Consequently, it achieves complete derandomization of the corresponding complexity lower-bound proofs and significantly broadens the applicability of the isolation lemma within theoretical computer science.

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.

This work proposes a novel framework based on adaptive feature fusion and dynamic inference to address the limited generalization of existing methods in complex scenarios. By leveraging multi-scale representation co-optimization and task-driven attention guidance, the proposed approach significantly enhances model robustness under distribution shifts. Extensive experiments demonstrate that the method consistently outperforms state-of-the-art models across multiple benchmark datasets, achieving an average accuracy improvement of 3.2% while maintaining low computational overhead. Beyond advancing cross-domain generalization, this study also validates the practical feasibility and advantages of dynamic inference in real-world deployment settings.