arXiv:2509.18057v5 Announce Type: replace 
Abstract: Can AI based methods help us make advances in complexity theory? We provide evidence towards answering this in the affirmative, using AlphaEvolve (an LLM code mutation agent) to obtain new results in three settings:
  a) 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 $163$ vertices, and our upper bounds are obtained via analytical arguments.
  b) We obtain new inapproximability results for MAX-4-CUT and MAX-3-CUT, proving that it is NP-hard to approximate them 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 the SOTA of $16/17$ that relies on a custom PCP (rather than a reduction from ``standard'' H{\aa}stad-style PCPs).
  c) Inapproximability for the metric Traveling Salesman Problem (TSP): We show that it is NP-hard to approximate the minimum cost tour within a factor of $111/110$ using AlphaEvolve to discover a new gadget, thus improving the SOTA of $117/116$. Along the way, we provide new modular soundness and completeness arguments that can be of independent interest.
  A key technical challenge we faced: verifying a candidate construction produced by AlphaEvolve is costly (sometimes requiring time exponential in the size of the construction). We used AlphaEvolve itself to evolve the verification procedure to be faster (sometimes by $10,000\times$ for our gadgets). Our results suggest that gadget based proofs would benefit from a pass through AI-based tools to obtain stronger results.

تم تحقيق تقدم حديث في نظرية التعقيد من خلال طرق تعتمد على الذكاء الاصطناعي، وخاصة باستخدام AlphaEvolve، وهو وكيل طفرات الشيفرة. وقد أدت هذه الأبحاث إلى تحسين الحدود العليا والسفلى على خوارزميات الشهادة لـ MAX-CUT و MAX-Independent Set على الرسوم البيانية العشوائية، بالإضافة إلى نتائج جديدة حول عدم القابلية للتقريب لـ MAX-4-CUT و MAX-3-CUT.

Se han logrado avances recientes en la teoría de la complejidad a través de métodos basados en IA, específicamente utilizando AlphaEvolve, un agente de mutación de código. Esta investigación ha llevado a mejorar los límites superiores e inferiores en los algoritmos de certificación para MAX-CUT y MAX-Independent Set en grafos aleatorios, así como a nuevos resultados de inaproximabilidad para MAX-4-CUT y MAX-3-CUT.

Des avancées récentes dans la théorie de la complexité ont été réalisées grâce à des méthodes basées sur l'IA, en particulier en utilisant AlphaEvolve, un agent de mutation de code. Cette recherche a conduit à des améliorations des bornes supérieures et inférieures sur les algorithmes de certification pour MAX-CUT et MAX-Independent Set sur des graphes aléatoires, ainsi qu'à de nouveaux résultats d'inapproximabilité pour MAX-4-CUT et MAX-3-CUT.

Recent advancements in complexity theory have been achieved through AI-based methods, specifically using AlphaEvolve, a code mutation agent. This research has led to improved upper and lower bounds on certification algorithms for MAX-CUT and MAX-Independent Set on random graphs, as well as new inapproximability results for MAX-4-CUT and MAX-3-CUT.

Reinforced Generation of Combinatorial Structures: Applications to Complexity Theory

<img src="https://res.infoq.com/news/2025/12/alphaevolve-google-cloud/en/headerimage/generatedHeaderImage-1765799824535.jpg"/><p>Google Cloud announced the private preview of AlphaEvolve, a Gemini-powered coding agent designed to discover and optimize algorithms for complex engineering and scientific problems. The system is now available through an early access program on Google Cloud, targeting use cases where traditional brute-force or manual optimization methods struggle due to vast search spaces.</p> <i>By Robert Krzaczyński</i>

أعلنت Google Cloud عن المعاينة الخاصة لـ AlphaEvolve، وهو وكيل برمجي مدعوم من Gemini مصمم لاكتشاف وتحسين الخوارزميات لمشاكل الهندسة والعلوم المعقدة. النظام متاح الآن من خلال برنامج الوصول المبكر، مستهدفًا بشكل خاص السيناريوهات التي تكون فيها طرق التحسين التقليدية غير كافية بسبب المساحات الواسعة للبحث.

Google Cloud ha anunciado la vista previa privada de AlphaEvolve, un agente de codificación impulsado por Gemini, diseñado para descubrir y optimizar algoritmos para problemas complejos de ingeniería y científicos. Este sistema ya está disponible a través de un programa de acceso anticipado, dirigido específicamente a escenarios donde los métodos de optimización tradicionales son inadecuados debido a amplios espacios de búsqueda.

Google Cloud a annoncé la prévisualisation privée d'AlphaEvolve, un agent de codage alimenté par Gemini, destiné à découvrir et optimiser des algorithmes pour des défis d'ingénierie et scientifiques complexes. Ce système est désormais disponible via un programme d'accès anticipé, ciblant spécifiquement les scénarios où les méthodes d'optimisation traditionnelles sont inadéquates en raison d'espaces de recherche étendus.

Google Cloud has announced the private preview of AlphaEvolve, a Gemini-powered coding agent aimed at discovering and optimizing algorithms for complex engineering and scientific challenges. This system is now available through an early access program, specifically targeting scenarios where traditional optimization methods are inadequate due to extensive search spaces.

AlphaEvolve Enters Google Cloud as an Agentic System for Algorithm Optimization

One More Thing in AI – Your Shortcut to AI Mastery

Reinforced Generation of Combinatorial Structures: Applications to Complexity Theory

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