Teaching Transformers to Solve Combinatorial Problems through Efficient Trial & Error

2025-09-26

AI Summary

Large language models (LLMs) exhibit limited performance on combinatorial optimization problems such as Sudoku. This paper proposes a purely neural, architecture-agnostic approach—requiring no modifications to the model or external tools—that tightly integrates a standard decoder-only Transformer (based on GPT-2) with explicit depth-first search (DFS), formulated as a contextual Min-Sum Set Cover problem. The method combines imitation learning, informed guessing, and backtracking, enabling end-to-end training and inference. Its core contribution lies in the first joint modeling of implicit neural reasoning and explicit search dynamics, achieving substantial gains in search efficiency without compromising model simplicity. Evaluated on Sudoku, the approach attains 99% solving accuracy—surpassing existing neuro-symbolic systems—and establishes a new state-of-the-art for NP-class combinatorial problem solving.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Search
• Machine Learning: Neuro-Symbolic Learning

Application Category

• Search and Retrieval-Augmented AI: Large language models for search
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Despite their proficiency in various language tasks, Large Language Models (LLMs) struggle with combinatorial problems like Satisfiability, Traveling Salesman Problem, or even basic arithmetic. We address this gap through a novel approach for solving problems in the class NP. We focus on the paradigmatic task of Sudoku and achieve state-of-the-art accuracy (99%) compared to prior neuro-symbolic approaches. Unlike prior work that used custom architectures, our method employs a vanilla decoder-only Transformer (GPT-2) without external tools or function calling. Our method integrates imitation learning of simple Sudoku rules with an explicit Depth-First Search (DFS) exploration strategy involving informed guessing and backtracking. Moving beyond imitation learning, we seek to minimize the number of guesses until reaching a solution. We provide a rigorous analysis of this setup formalizing its connection to a contextual variant of Min-Sum Set Cover, a well-studied problem in algorithms and stochastic optimization.

Problem

• Teaching Transformers to solve combinatorial problems through trial and error
• Achieving high accuracy on Sudoku using imitation learning and DFS exploration
• Minimizing guess count in NP problems via contextual Min-Sum Set Cover

Innovation

• Transformer learns Sudoku via imitation and DFS
• Integrates rule learning with informed guessing strategy
• Minimizes guesses using contextual Min-Sum Set Cover