The Art of Being Difficult: Combining Human and AI Strengths to Find Adversarial Instances for Heuristics

2026-01-23

AI Summary

This work addresses the theoretical limitations of classical heuristics by constructing adversarial instances that significantly degrade their performance. Leveraging the FunSearch framework, we iteratively refine initial solutions through a synergistic integration of human mathematical insight and the evolutionary generation capabilities of large language models (LLMs). Combined with manual fine-tuning and formal verification, this approach yields breakthroughs on several long-standing combinatorial optimization problems that had seen no progress for over a decade. Specifically, our method establishes new lower bounds—surpassing records held for more than ten years—for generalized variants of hierarchical k-median clustering, bin packing, knapsack, and the Lovász gasoline problem, thereby demonstrating the transformative potential of human–AI collaboration in theoretical computer science.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Search
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

We demonstrate the power of human-LLM collaboration in tackling open problems in theoretical computer science. Focusing on combinatorial optimization, we refine outputs from the FunSearch algorithm [Romera-Paredes et al., Nature 2023] to derive state-of-the-art lower bounds for standard heuristics. Specifically, we target the generation of adversarial instances where these heuristics perform poorly. By iterating on FunSearch's outputs, we identify improved constructions for hierarchical $k$-median clustering, bin packing, the knapsack problem, and a generalization of Lov\'asz's gasoline problem - some of these have not seen much improvement for over a decade, despite intermittent attention. These results illustrate how expert oversight can effectively extrapolate algorithmic insights from LLM-based evolutionary methods to break long-standing barriers. Our findings demonstrate that while LLMs provide critical initial patterns, human expertise is essential for transforming these patterns into mathematically rigorous and insightful constructions. This work highlights that LLMs are a strong collaborative tool in mathematics and computer science research.

Problem

• adversarial instances
• combinatorial optimization
• heuristics
• lower bounds
• theoretical computer science

Innovation

• human-AI collaboration
• adversarial instances
• combinatorial optimization
• FunSearch
• heuristics