combinatorial reasoning

This paper investigates conditional lower bounds for the evaluation complexity of conjunctive queries (CQs) under Boolean testing, counting, enumeration, and direct-access computation models. To address fundamental performance bottlenecks resistant to existing algorithms, the authors employ fine-grained complexity assumptions—including 3SUM and APSP—and integrate parameterized analysis with conditional lower-bound proof techniques. They establish the first tight conditional lower bounds for multiple CQ variants, demonstrating that, under standard hypotheses, current state-of-the-art algorithms are asymptotically optimal—any further improvement faces inherent barriers. The work systematically characterizes the complexity landscape across diverse query semantics and computational models, thereby providing rigorous theoretical limits for query optimization. These results significantly advance the understanding of the intrinsic feasibility and hardness of relational query processing.

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.

This paper investigates the computational power and complexity boundaries of families of nonuniform polynomial-size nondeterministic finite automata (NFA-poly) with respect to partial counting functions, gap functions, and their associated promise decision problems. Using state-complexity analysis and counting-complexity frameworks, it provides the first systematic characterization of the complexity hierarchy for NFA-poly without relying on unproven hardness assumptions. Key contributions include: (i) a strict separation between the counting class #NFA-poly and the gap class GapNFA-poly; (ii) an exact simulation equivalence between these classes and polynomially stack-bounded pushdown automata (PDA); and (iii) identification of novel conditions under which complexity collapses occur for promise problems. These results unify the theoretical connections between counting-based finite automata and resource-restricted PDAs, yielding a foundational stratification of automata-based complexity classes.

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 paper addresses the computational challenge of efficiently evaluating the polymatroid bound under simplicity constraints in cardinality estimation and conjunctive query evaluation. We present the first polynomial-time exact algorithm for this problem. Our contributions are threefold: (1) the first $O( ext{poly}(n))$ exact algorithm for the polymatroid bound under simplicity constraints; (2) a novel flow bound that improves computational efficiency while preserving tightness and scalability; and (3) generation of verifiable proof sequences, seamlessly integrated into the PANDA framework to accelerate query evaluation. Theoretically, we establish computational intractability for extending our method to general constraint classes. Experimental results demonstrate substantial improvements in conjunctive query evaluation performance, establishing a new paradigm for information-theoretic cardinality estimation that bridges theoretical rigor with practical deployability.

This work investigates the non-clashing teaching problem for concept classes defined by closed neighborhoods in graphs—a batch teaching model that satisfies anti-collusion criteria but suffers from high computational complexity and is only solvable on restricted graph classes. By leveraging parameterized complexity analysis, combinatorial graph-theoretic techniques, and reduction methods, we design the first fixed-parameter tractable (FPT) algorithm applicable to a significantly broader family of graphs. Our main contributions include an improved FPT algorithm for general graphs, tight combinatorial upper bounds, and proofs of strong inapproximability and W[1]-hardness results. These advances substantially deepen the understanding of both the algorithmic and complexity-theoretic landscape of this teaching problem.

This work investigates how to achieve approximately uniform edge sampling in sublinear time under a hybrid query model that combines independent set queries and local graph queries. It establishes, for the first time, a tight two-way reduction between edge sampling and approximate edge counting in this model, demonstrating that the two problems share matching upper and lower bounds in query complexity. The proposed sampling algorithm matches the query complexity of the current best-known edge counting algorithms, while the analysis yields tight lower bounds for each constituent query model. These results highlight the pivotal role of independent set queries in enabling efficient graph sampling tasks, revealing a fundamental equivalence between sampling and counting in terms of computational hardness under the hybrid query framework.

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 investigates lower bounds on the query complexity of Boolean functions in the noisy query model, where each query is flipped independently with a fixed probability. By analyzing degree statistics of specific subgraphs of the Boolean hypercube and integrating tools from Boolean function analysis, hypercube graph theory, and probabilistic methods, the authors introduce the first general lower bound framework that surpasses the trivial random-query baseline. This framework not only unifies and simplifies existing results but also resolves an open problem concerning the relationship between total influence $I(f)$ and noisy query complexity $N_p(f)$, establishing that $N_p(f) = \Omega(I(f) \log I(f))$. Furthermore, it yields tight bounds for several new functions and recovers nearly all known lower bounds up to constant factors.

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.