Reinforced Generation of Combinatorial Structures: Applications to Complexity Theory

We explore whether techniques from AI can help discover new combinatorial structures that improve on known 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" H{\aa}stad-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 $10,000\times$). We conclude with a discussion of norms by which to assess the assistance from AI in developing proofs.