exact enumeration

This study systematically evaluates the rigorous proof reasoning and explicit construction capabilities of large language models on Olympiad-level combinatorics problems. To this end, we introduce a benchmark comprising 100 expert-annotated competition problems, categorizing tasks into analytical (proof-oriented) and constructive (implementation-oriented) types. We propose a unified evaluation protocol that integrates rubric-guided proof assessment with deterministic verification of constructions, enhanced by a Best@4 multi-solution sampling strategy. Experimental results show that the strongest model achieves an average score of 65.4% overall (75.3% under Best@4), with markedly divergent performance across the two task types, revealing current limitations in creative mathematical reasoning—particularly on existence and construction problems. This work presents the first fine-grained distinction and joint evaluation of these capabilities, offering a new benchmark and diagnostic framework for mathematical reasoning research.

Large language models (LLMs) lack standardized, formalized evaluation benchmarks for combinatorial mathematics. Method: We introduce CombiBench—the first Lean 4–based formal benchmark for combinatorics—comprising 100 proof and fill-in-the-blank problems spanning primary to International Mathematical Olympiad (IMO) difficulty levels across十余 combinatorial topics; we further propose Fine-Eval, the first automated framework enabling precise scoring of formalized fill-in-the-blank problems, and comprehensively formalize all non-geometric IMO combinatorics problems since 2000. Contribution/Results: Experiments reveal severely limited zero-shot formal solving capabilities of current LLMs (maximum 7/100 solved), underscoring the profound challenge of formal combinatorial reasoning. CombiBench fills a critical gap in the field, providing a reproducible, extensible benchmark and evaluation infrastructure to advance research in formal mathematical reasoning.

This study addresses the construction of functions in algebraic combinatorics subject to stringent distributional constraints and the discovery of previously unknown combinatorial symmetries. To this end, we propose the SLURP framework, which integrates MapSeek-Functional and MapSeek-Symbolic approaches through alternating pseudo-label supervised learning, symbolic regression, and formal verification in Lean 4. The framework yields the first combinatorial interpretation of $q,t$-Narayana polynomials based on non-crossing partitions and provides a combinatorial proof of symmetry in previously unresolved cases by leveraging newly discovered statistics. All code and formalized results are publicly released to ensure reproducibility and rigorous verification.

ApproxMC, a hashing-based approximate model counter, suffers from suboptimal internal parameter configurations that hinder its practical scalability. Method: We propose the first parameter optimization framework for ApproxMC with PAC-correctness guarantees, formalizing parameter selection as a decoupled optimization problem—separating correctness verification from performance maximization—and deriving a minimal, efficiently searchable parameter expression. Our approach integrates PAC learning theory, hash function design, and constrained optimization modeling to maximize runtime efficiency while provably ensuring (ε, δ)-approximation accuracy. Contribution/Results: Experimental evaluation demonstrates that our method accelerates the latest ApproxMC by 1.6–2.4× across diverse ε-tolerance settings, significantly improving real-world scalability without compromising theoretical correctness guarantees.

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 challenges in combinatorial counting arising from intricate structural and arithmetic constraints, which hinder manual derivation and cause existing methods to break problem symmetries. To overcome these limitations, the paper introduces Cofola, a typed declarative language that unifies combinatorial counting as a weighted first-order model counting (WFOMC) problem with coefficient extraction constraints—the first such formulation. Cofola naturally expresses common combinatorial structures including sets, multisets, permutations, and partitions. Its three-stage compilation pipeline integrates preprocessing, symmetry-preserving decomposition, and ordering axiom encoding—such as lexicographic symmetry breaking and sequence/cycle axioms—to enable efficient solving while preserving inherent symmetries. Experimental results demonstrate that Cofola substantially outperforms existing frameworks in both expressiveness and computational efficiency across a diverse benchmark suite, ranging from textbook examples to complex multi-object scenarios.

This work proposes a novel Monte Carlo approximation algorithm for counting models in disjunctive normal form (DNF), a problem of central importance in probabilistic inference and network reliability that is notoriously intractable due to its high computational complexity. The method uniquely integrates adaptive stopping rules with short-circuit formula evaluation, achieving substantial gains in sampling efficiency while preserving theoretical guarantees under the Probably Approximately Correct (PAC) framework. It is both asymptotically optimal and practically scalable, as demonstrated by extensive experiments showing speedups of several orders of magnitude over state-of-the-art approaches. Notably, the algorithm efficiently handles DNF instances with up to millions of variables, significantly advancing the frontier of feasible model counting in large-scale settings.

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 investigates, under the assumption that P ≠ NP, whether NP-complete problems admit algorithms substantially faster than naïve brute-force search and whether current best-known algorithms are already optimal. By integrating fine-grained complexity theory, algebraic techniques, extremal and additive combinatorics, cryptography, and conditional hypotheses such as the Strong Exponential Time Hypothesis (SETH), the project establishes a unified framework for deriving conditional time lower bounds for NP-complete problems. Through a systematic synthesis of classical and recent results, and by leveraging reductions and combinatorial analyses, the study provides strong evidence for the hardness of improving existing algorithms for several canonical NP-complete problems, thereby advancing our understanding of the fine-grained structure of computational complexity.

This work addresses the exponential explosion in enumerating unique paths in recombining trinomial trees—traditionally requiring $mathcal{O}(3^D)$ operations. We propose a *mass-sliding enumeration algorithm* leveraging translational invariance of nodes under time-homogeneous dynamics and a weakly combinatorial bijective encoding. The method uniquely generates one canonical representative per equivalence class of paths while enabling implicit counting. It establishes tight combinatorial upper bounds and provable algorithmic lower complexity bounds, uncovering deep connections to Motzkin paths and Narayana-type structures. By exploiting ordered-tuple mapping, graph symmetry analysis, and depth-first traversal—without explicit path enumeration—the algorithm achieves negligible constant overhead. Experiments demonstrate substantial speedup over breadth-first enumeration, enabling exact path enumeration and high-precision numerical approximation of evolutionary processes in deeper trees.