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

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 ($C \ge 2.2203$). In information theory, we uncover a larger independent set for the 15-vertex cycle graph ($\mathcal{C}_{15}^{\boxtimes 5}$, 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.