# (X)-evolve: Solution space evolution powered by large language models

arXiv: https://arxiv.org/abs/2508.07932 · HTML: https://mlsanity.com/paper/2508.07932/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 98 references that are in the map.
- First posted on arXiv 11 Aug 2025.

## Builds on (most cited first, 30 of 46)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [SWE-bench: Can Language Models Resolve Real-World GitHub Issues?](https://mlsanity.com/paper/2310.06770/) (2023)
- [RT-2: Vision-Language-Action Models Transfer Web Knowledge to Robotic Control](https://mlsanity.com/paper/2307.15818/) (2023)
- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [Do As I Can, Not As I Say: Grounding Language in Robotic Affordances](https://mlsanity.com/paper/2204.01691/) (2022)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Code Llama: Open Foundation Models for Code](https://mlsanity.com/paper/2308.12950/) (2023)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019)
- [Qwen2.5-Coder Technical Report](https://mlsanity.com/paper/2409.12186/) (2024)
- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Evolution Strategies as a Scalable Alternative to Reinforcement Learning](https://mlsanity.com/paper/1703.03864/) (2017)
- [Eureka: Human-Level Reward Design via Coding Large Language Models](https://mlsanity.com/paper/2310.12931/) (2023)
- [Promptbreeder: Self-Referential Self-Improvement Via Prompt Evolution](https://mlsanity.com/paper/2309.16797/) (2023)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Describe, Explain, Plan and Select: Interactive Planning with Large Language Models Enables Open-World Multi-Task Agents](https://mlsanity.com/paper/2302.01560/) (2023)
- [DeepSeek-Coder-V2: Breaking the Barrier of Closed-Source Models in Code Intelligence](https://mlsanity.com/paper/2406.11931/) (2024)
- [Evolution of Heuristics: Towards Efficient Automatic Algorithm Design Using Large Language Model](https://mlsanity.com/paper/2401.02051/) (2024)
- [Reward Design with Language Models](https://mlsanity.com/paper/2303.00001/) (2023)
- [KernelBench: Can LLMs Write Efficient GPU Kernels?](https://mlsanity.com/paper/2502.10517/) (2025)

## Built on by (most cited first, 6 of 6)

- [The FM Agent](https://mlsanity.com/paper/2510.26144/) (2025)
- [Glia: A Human-Inspired AI for Automated Systems Design and Optimization](https://mlsanity.com/paper/2510.27176/) (2025)
- [What Do Evolutionary Coding Agents Evolve?](https://mlsanity.com/paper/2605.20086/) (2026)
- [Game-Theoretic Co-Evolution for LLM-Based Heuristic Discovery](https://mlsanity.com/paper/2601.22896/) (2026)
- [HybridQuestion: Human-AI Collaboration for Identifying High-Impact Research Questions](https://mlsanity.com/paper/2602.03849/) (2025)
- [Improved lower bounds for the Shannon capacity of odd cycles](https://mlsanity.com/paper/2607.21517/) (2026)

## Abstract (arXiv)

While combining large language models (LLMs) with evolutionary algorithms (EAs) shows promise for solving complex optimization problems, current approaches typically evolve individual solutions, often incurring high LLM call costs. We introduce (X)-evolve, a paradigm-shifting method that instead evolves solution spaces (X) (sets of individual solutions) - subsets of the overall search space (S). In (X)-evolve, LLMs generate tunable programs wherein certain code snippets, designated as parameters, define a tunable solution space. A score-based search algorithm then efficiently explores this parametrically defined space, guided by feedback from objective function scores. This strategy enables broader and more efficient exploration, which can potentially accelerate convergence at a much lower search cost, requiring up to two orders of magnitude fewer LLM calls than prior leading methods. We demonstrate (X)-evolve's efficacy across three distinct hard optimization problems. For the cap set problem, we discover a larger partial admissible set, establishing a new tighter asymptotic lower bound for the cap set constant ((C ≥ 2.2203)). In information theory, we uncover a larger independent set for the 15-vertex cycle graph ((C15boxtimes 5), size 19,946), thereby raising the known lower bound on its Shannon capacity. Furthermore, for the NP-hard online bin packing problem, we generate heuristics that consistently outperform standard strategies across established benchmarks. By evolving solution spaces, our method considerably improves search effectiveness, making it possible to tackle high-dimensional problems that were previously computationally prohibitive.