combinatorial design search

This work addresses long-standing existence problems in combinatorial design—particularly those unresolved or unconstructed in the *Handbook of Combinatorial Designs*. We propose CPro1, the first framework integrating large language model (LLM)-driven code generation with formally specified, verifiable construction definitions and an automated feedback loop, enabling fully automated design search and hyperparameter optimization without manual coding. Our method combines domain-specific validators, a scaffolding-based execution framework, and heuristic optimization algorithms—including simulated annealing and genetic algorithms—to systematically explore construction strategies. Evaluated on 16 classical combinatorial design problems, CPro1 successfully resolves six open instances: symmetric and skew weighing matrices, isometric arrays, packing arrays, balanced ternary designs, and Florentine rectangles—marking a significant advance beyond traditional hand-crafted construction methods.

Many existence problems in combinatorial design remain open, lacking constructive solutions or effective search heuristics. Method: We propose a novel framework that integrates reasoning-oriented large language models (LLMs) into the constructive solving protocol CPro1, enabling end-to-end generation of executable search heuristics directly from problem specifications. Our approach unifies LLM-driven code generation, automated correctness verification, hyperparameter optimization, and execution feedback in a closed loop. Contribution/Results: Applied to 16 long-standing open instances from the *Handbook of Combinatorial Designs* (2006), our method successfully constructs solutions for 7 cases—including three problem classes resolved for the first time. Moreover, it discovers several new combinatorial structures recently reported in 2025 literature. By automating heuristic discovery and validation, this work substantially advances the frontier of constructive combinatorial design automation.

This work addresses permutation pattern avoidance and containment via a declarative constraint programming (CP) approach. Methodologically, it introduces a composable and extensible library of permutation constraints, enabling unified modeling of six pattern avoidance/containment relations, eight classical permutation properties, and five combinatorial statistics; it further achieves the first dynamic composition and incremental solving of arbitrary pattern constraints. As a key application, the framework enumerates inversions in 1324-avoiding permutations—extending the tractable instance size to length 16 for the first time—and discovers that the resulting inversion distribution corresponds to a novel integer sequence not yet cataloged in OEIS. Experiments demonstrate substantial improvements in both efficiency and flexibility for generating and analyzing permutations under complex combinatorial constraints. The proposed framework establishes a reusable, declarative paradigm for enumerative combinatorics, bridging CP methodology with structural enumeration problems.

The computational complexity of graph isomorphism testing remains unresolved, particularly for highly symmetric graphs whose adjacency matrices possess repeated eigenvalues—causing ambiguity in the solution space for conventional methods. This paper introduces a novel continuous optimization framework: it reformulates the discrete matching problem via orthogonal and doubly stochastic relaxations, and—crucially—systematically characterizes how eigenvalue multiplicity governs the geometric structure of the feasible solution space. Building on this insight, we propose a subspace constraint strategy that effectively suppresses spurious solutions induced by symmetry. Our algorithm employs the Frank–Wolfe method, integrating spectral matrix analysis with iterative projection-based optimization. Extensive evaluation on highly symmetric benchmarks—including strongly regular graphs, complete graphs, and the Petersen graph—demonstrates substantial improvements in both efficiency and robustness of isomorphism detection.

This work addresses the satisfiability problem for constraint satisfaction problems (CSPs) over homogeneous infinite hypergraphs, circumventing the fundamental limitation that finite-domain reduction techniques do not directly extend to infinite domains. We devise the first symmetry-driven algorithm for this setting—relying on an external linear order—and integrate tools from homogeneous structure model theory, first-order definability reductions, and group action analysis to systematically classify the computational complexity of broad classes of infinite hypergraph CSPs. Our main contribution is the establishment of a dichotomy theorem—classifying each CSP as either in P or NP-complete—for a wide family of infinite hypergraphs. This result confirms the Bodirsky–Pinsker conjecture in full generality and significantly extends prior graph-CSP dichotomies to higher-arity hypergraph structures. The framework provides a unified classification theory and foundational technical machinery for infinite-domain CSPs.

This work addresses the limitation in combinatorial optimization where local search neighborhoods typically require manual construction. It proposes, for the first time, a method that automatically generates functional neighborhoods by exploiting symmetries present in constraint specifications. By integrating constraint programming, symmetry analysis, and local search techniques, the approach enables automated neighborhood construction within the IDP system, substantially reducing the need for human intervention. Empirical evaluation across six classical optimization problems demonstrates the effectiveness of the generated neighborhoods, confirming both the feasibility of the method and its capacity to enhance the automation and generality of local search algorithms.

This work addresses the Bandwidth Coloring Problem (BCP), which imposes a minimum separation constraint on colors assigned to adjacent vertices and has important applications in domains such as frequency assignment. The study presents a systematic investigation of SAT-based encodings for BCP, introducing a unified framework that encompasses single-variable, double-variable, and block encodings. For the first time in this context, symmetry-breaking constraints and an incremental solving strategy are incorporated. The proposed block encoding substantially enhances solver efficiency, achieving state-of-the-art performance on the GEOM and MS-CAP benchmarks. Notably, it proves the optimality of the GEOM120b instance within approximately 1,000 seconds—a result unattainable by prior methods even after one hour of computation.

This study investigates the solvability and repair of permutation-matching puzzles on an $n \times n$ grid subject to row- and column-wise sorting constraints (either ascending or descending). By constructing a constraint graph, the work provides the first complete characterization of solvability conditions, introducing a “at most one switch” criterion to determine the existence of a solution. For solvable instances, it presents a counting method based on the hook-length formula; for unsolvable ones, it designs a linear-time algorithm to compute the minimum number of label flips required for repair. The framework is further extended to arbitrary permutation constraints, where the minimum repair problem is shown to be NP-complete. Integrating combinatorics, graph theory, and computational complexity, this work establishes a theoretical foundation and efficient algorithmic tools for this class of puzzles.

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.