QUBE: Enhancing Automatic Heuristic Design via Quality-Uncertainty Balanced Evolution

TL;DR

QUBE addresses the challenge of automatically designing heuristics for NP-hard problems by rebalancing exploitation and exploration in LLM+EA methods through the Quality-Uncertainty Trade-off Criterion (QUTC) and the Uncertainty-Inclusive Quality (UIQ) metric. It redefines the evolution priorities to operate at the cluster level, using UIQ for cluster selection and an island-reset strategy to maintain diversity while concentrating search on promising regions. Empirical results across online bin packing, cap set, and traveling salesman problem show significant improvements over FunSearch and several baselines, with robust performance across different LLMs and settings; cap set results are strong but not strictly surpassing FunSearch in all reproduced scenarios. The work highlights a scalable, data-driven pathway to enhance heuristic discovery and provides practical code for reproducibility. All mathematical formulations are presented with proper Delimiters, including $\tilde{Q}_t(\mathbb{C})=Q_t(\mathbb{C})+k\sqrt{\frac{\ln t}{N_t(\mathbb{C})}}$ and related terms.

Abstract

Solving NP-hard problems traditionally relies on heuristics, yet manually designing effective heuristics for complex problems remains a significant challenge. While recent advancements like FunSearch have shown that large language models (LLMs) can be integrated into evolutionary algorithms (EAs) for heuristic design, their potential is hindered by limitations in balancing exploitation and exploration. We introduce Quality-Uncertainty Balanced Evolution (QUBE), a novel approach that enhances LLM+EA methods by redefining the priority criterion within the FunSearch framework. QUBE employs the Quality-Uncertainty Trade-off Criterion (QUTC), based on our proposed Uncertainty-Inclusive Quality metric, to evaluate and guide the evolutionary process. Through extensive experiments on challenging NP-complete problems, QUBE demonstrates significant performance improvements over FunSearch and baseline methods. Our code are available at https://github.com/zzjchen/QUBE_code.

Figures (6)

• Figure 1: Experiment results of our method on online bin packing, the performance is evaluated with "Excess Ratio" and the lower the better. Our method can steadily find better heuristics than all baselines.
• Figure 2: Illustration of QUBE. We manipulate the parent selection procedure in FunSearch's evolution process. Left: The overall evolution process of our method and FunSearch. Right: At each timestep, FunSearch selects parents based on the score of each sample. Our method selects parents based on our quality measure, UIQ. The uncertainty of a sample's quality is acquired from the number of times it is selected as parents.
• Figure 3: The "Recent Best Score" of FunSearch exhibits plateaus in later stages, indicating challenges in effectively exploiting known regions. In contrast, our method consistently generates higher-scoring samples, demonstrating superior exploitation capabilities.
• Figure 4: FunSearch has a consistently low "Recent Proportion of Change", reflecting its limited overall exploration of the function space. In contrast, our method demonstrates both a broader scope and a more intelligent exploration strategy, enabling more effective discovery of promising regions.
• Figure 5: Performance progress on online bin packing. The solid line shows the average score among 10 experiments at each timestep. The shadow shows the range of best and worst experiments. FunSearch is shown in dash line since only a final score is available.