Self-Guiding Exploration for Combinatorial Problems

TL;DR

The paper tackles the challenge of using large language models to solve combinatorial problems, which are NP-hard and central to logistics and scheduling. It introduces Self-Guiding Exploration (SGE), a general-purpose prompting framework that autonomously generates multiple thought trajectories, decomposes them into subtasks, executes and refines them, and then integrates them into a final solution. Across six CP tasks and various problem sizes, SGE outperforms traditional prompting strategies (CoT, Decomposition, Refine) with a 27.84% improvement in CP optimization and achieves a 2.46% higher accuracy on broader reasoning tasks, with larger gains on more complex problems. The work demonstrates the potential of advanced, autonomous LLM prompting to enhance performance in critical optimization domains, while noting computational cost and model-dependence as areas for future improvement.

Abstract

Large Language Models (LLMs) have become pivotal in addressing reasoning tasks across diverse domains, including arithmetic, commonsense, and symbolic reasoning. They utilize prompting techniques such as Exploration-of-Thought, Decomposition, and Refinement to effectively navigate and solve intricate tasks. Despite these advancements, the application of LLMs to Combinatorial Problems (CPs), known for their NP-hardness and critical roles in logistics and resource management remains underexplored. To address this gap, we introduce a novel prompting strategy: Self-Guiding Exploration (SGE), designed to enhance the performance of solving CPs. SGE operates autonomously, generating multiple thought trajectories for each CP task. It then breaks these trajectories down into actionable subtasks, executes them sequentially, and refines the results to ensure optimal outcomes. We present our research as the first to apply LLMs to a broad range of CPs and demonstrate that SGE outperforms existing prompting strategies by over 27.84% in CP optimization performance. Additionally, SGE achieves a 2.46% higher accuracy over the best existing results in other reasoning tasks (arithmetic, commonsense, and symbolic).

Figures (4)

• Figure 1: Self-Guiding Exploration. The generative model autonomously addresses a combinatorial problem task $Q$ through a five-phase process: (1) Exploration of $N$ solution trajectories, where each trajectory offers potential solutions; (2) Decomposition of these trajectories into $K$ subtasks, outlining specific steps for each method; (3) Resolution of each subtask, executing the outlined steps; (4) Feedback and Refinement, where feedback is gathered and used to refine each subtask; (5) Integration of all trajectories into a consolidated final solution $A$. Distinct from traditional exploration/decomposition techniques, SGE(Q) functions entirely autonomously, eliminating the reliance on task-specific queries or manually created thought exemplars. This independence makes it universally applicable to all CP tasks without necessitating modifications.
• Figure 2: Example of SGE inference across the Exploration, Decomposition, and Refinement phases for the Traveling Salesman Problem. The figure displays three boxes, each illustrating the prompt structure and corresponding example output for each phase.
• Figure 3: Effect of Problem Size on Performance Improvement relative to the IO solution using gpt-4 w/ code interpreter. The analysis spans problem instances of varying sizes, systematically presented from the smallest to the largest, specifically ranging from $n=5$ to $n=20$ nodes. Results are organized to highlight the impact of increasing problem complexity on the effectiveness of the solution.
• Figure 4: Effect of Model Choice on Performance Improvement relative to the IO solution.