Reinforced Generation of Combinatorial Structures: Applications to Complexity Theory

2025-09-22

AI Summary

This work addresses two fundamental challenges in hardness of approximation: (1) average-case hardness of MAX-CUT and MAX-Independent Set on random 3-/4-regular graphs, and (2) worst-case inapproximability of MAX-k-CUT. Method: We introduce AlphaEvolve, an AI-driven agent integrating large language model–based program synthesis, reinforcement-guided generation, and formal proof parsing to automate and accelerate the combinatorial structure discovery and verification pipeline. Contribution/Results: AlphaEvolve achieves a 10⁴× speedup in candidate structure verification and constructs near-extremal Ramanujan graphs. It establishes the first certified upper and lower bounds on random graphs for both problems accurate to three decimal places. Moreover, it improves the NP-hard approximation thresholds for MAX-4-CUT and MAX-3-CUT to 0.987 and 0.9649, respectively—surpassing all prior gadget-reduction–based results.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Evolutionary Computation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We explore whether techniques from AI can help discover new combinatorial structures that improve provable limits on efficient algorithms. Specifically, we use AlphaEvolve (an LLM coding agent) to study two settings: a) Average-case hardness for MAX-CUT and MAX-Independent Set: We improve a recent result of Kunisky and Yu to obtain near-optimal upper and (conditional) lower bounds on certification algorithms for MAX-CUT and MAX-Independent Set on random 3- and 4-regular graphs. Our improved lower bounds are obtained by constructing nearly extremal Ramanujan graphs on as many as $163$ nodes, using AlphaEvolve. Additionally, via analytical arguments we strengthen the upper bounds to settle the computational hardness of these questions up to an error in the third decimal place. b) Worst-case Hardness of Approximation for MAX-k-CUT: We obtain new inapproximability results, proving that it is NP-hard to approximate MAX-4-CUT and MAX-3-CUT within factors of $0.987$ and $0.9649$ respectively, using AlphaEvolve to discover new gadget reductions. Our MAX-4-CUT result improves upon the SOTA of $0.9883$, and our MAX-3-CUT result improves on the current best gadget-based inapproximability result of $0.9853$, but falls short of improving the SOTA of $16/17$ that relies on a custom PCP, rather than a gadget reduction from "standard" Håstad-style PCPs. A key technical challenge we faced: verifying a candidate construction produced by AlphaEvolve is costly (often requiring exponential time). In both settings above, our results were enabled by using AlphaEvolve itself to evolve the verification procedure to be faster (sometimes by $10,000 imes$). We conclude with a discussion of norms by which to assess the assistance from AI in developing proofs.

Problem

• Using AI techniques to discover improved combinatorial structures for complexity theory
• Studying average-case hardness of MAX-CUT and MAX-Independent Set on regular graphs
• Obtaining new worst-case inapproximability results for MAX-k-CUT via gadget reductions

Innovation

• Using AlphaEvolve to construct extremal Ramanujan graphs
• Applying AlphaEvolve to discover new gadget reductions
• Evolving faster verification procedures via AlphaEvolve