Using Reasoning Models to Generate Search Heuristics that Solve Open Instances of Combinatorial Design Problems

TL;DR

This paper introduces CPro1, a reasoning-based approach using Large Language Models to generate search heuristics that successfully solve several long-standing open combinatorial design problems, including new solutions.

Contribution

The paper presents a novel method that leverages reasoning LLMs to create effective search heuristics for solving open instances in combinatorial design, outperforming non-reasoning models.

Findings

Successfully solved 7 of 16 open problems from the 2006 Handbook.

Generated new solutions for 3 previously unsolved design types.

Extended the approach to recent literature problems, producing new combinatorial structures.

Abstract

Large Language Models (LLMs) with reasoning are trained to iteratively generate and refine their answers before finalizing them, which can help with applications to mathematics and code generation. We apply code generation with reasoning LLMs to a specific task in the mathematical field of combinatorial design. This field studies diverse types of combinatorial designs, many of which have lists of open instances for which existence has not yet been determined. The Constructive Protocol CPro1 uses LLMs to generate search heuristics that have the potential to construct solutions to small open instances. Starting with a textual definition and a validity verifier for a particular type of design, CPro1 guides LLMs to select and implement strategies, while providing automated hyperparameter tuning and execution feedback. CPro1 with reasoning LLMs successfully solves long-standing open instances for 7 of 16 combinatorial design problems selected from the 2006 Handbook of Combinatorial Designs, including new solved instances for 3 of these (Bhaskar Rao Designs, Symmetric Weighing Matrices, Balanced Ternary Designs) that were unsolved by CPro1 with non-reasoning LLMs. It also solves open instances for several problems from recent (2025) literature, generating new Covering Sequences, Johnson Clique Covers, Deletion Codes, and a Uniform Nested Steiner Quadruple System.