LLM-Guided Program Evolution for Circle Packing: Breaking 10 Packomania Records for $28

Wes Sander

cs.AISep 4, 2026

LLM-Guided Program Evolution for Circle Packing: Breaking 10 Packomania Records for $28

Authors: Wes Sander

Organizations: Practical Systems Mason, Ohio, USA

Abstract

We present Discovery Loop, a lightweight system that uses a large language model (LLM) to iteratively evolve optimization algorithms. Starting from a simple seed solver, the LLM proposes algorithmic improvements guided by a scoreboard of results and a history of prior ideas. Each candidate is evaluated against an independent verifier; improvements are kept and failures discarded. Applied to the Packomania circle-packing benchmark (csqv: maximize the sum of radii of N variable-radius circles in the unit square), the system improved the best known solutions for 10 values of N in the range 101-114, with gains of 2.4%-5.4% over prior records, all within 15 iterations and at a total LLM cost of $27.72. These results have been independently accepted by Packomania. We describe the method, analyze cost-efficiency dynamics including an adaptive plateau-detection mechanism, and discuss implications for democratizing automated scientific discovery.

Figures & tables

$N$	Prior Record	Our Best	Improvement	First Improved
101	5.163845	5.289154	+2.43%	Iter 0 (seed)
102	5.055187	5.318238	+5.20%	Iter 0 (seed)
103	5.085509	5.345481	+5.11%	Iter 0 (seed)
105	5.125967	5.401298	+5.37%	Iter 0 (seed)
106	5.151736	5.429079	+5.38%	Iter 0 (seed)
107	5.180124	5.453952	+5.29%	Iter 0 (seed)

Table 1: Improvements over prior Packomania records (csqv). All results independently verified with zero tolerance and accepted by Packomania.

Iter	Status	Cost ($)	Total	Key Innovation
0	seed	0.00	59.39	Penalty L-BFGS-B + LP radii
1	champion	1.19	59.49	Basin hopping + SLSQP contact polish
2	champion	1.05	59.66	Sparse penalty + adaptive operators + elite pool
3	no-code	0.00	—	(Failed output)
4	champion	1.00	59.90	Hexagonal lattice init + crossover
5	champion	1.72	59.96	Island-model parallel basin hopping

Table 2: Iteration log. Total is

\sum_{t}\text{score}(v_{t},R_{t})

across all 12 targets.

	AlphaEvolve	FunSearch	Discovery Loop
Organization	Google DeepMind	Google DeepMind	Individual
Infrastructure	Distributed cluster	Distributed cluster	Single consumer PC
Evaluation	Distributed sandbox	Distributed sandbox	Local parallel
LLM calls/iter	Multiple	Multiple	1
Code granularity	Function patches	Small functions	Full program replacement
Problem domains	Multiple	Constructive math	Optimization

Table 3: Comparison with LLM-guided program evolution systems.

Automated Algorithm Discovery

Automated algorithm discovery develops methods that generate and test executable algorithms for optimization, science, software, and machine learning.

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Sep 30, 2026Mansi Sakarvadia, Marco Ciccone, Colin RaffelLarge Language Model-Guided Optimization Large Language Model-Guided Scientific Discovery
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Feb 10, 2026J/K move · Enter open · S save

ImprovEvolve: Basin-Hopping Meets LLM-Guided Evolutionary Search

Alexey Kravatskiy, Valentin Khrulkov, Ivan Oseledets

1MIRIAI, Russia · 2FusionBrain Lab, Russia · 3AXXX, Russia+1

Large Language Model-Guided Scientific Discovery Evolutionary Optimization Metaheuristic Optimization Large Language Model-Guided Optimization Combinatorial Optimization Global Optimization+2

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LLM-guided evolutionary computation, most notably AlphaEvolve, has been remarkably successful in discovering novel mathematical constructions by solving challenging optimization problems. The standard approach is to evolve a monolithic program that directly outputs a candidate solution. We present ImprovEvolve, an algorithmic alternative that drastically reduces cognitive load on the LLM. Instead of prompting the model for an end-to-end optimizer, we evolve a program with three specialized operators of initialization, local improvement, and perturbation. We then approach the optimum by iteratively applying local improvements and intensity-scheduled perturbations, effectively driving a basin-hopping search with LLM-evolved subroutines. For hexagon in hexagon packing, ImprovEvolve discovers new state-of-the-art packings of 11, 12, 15, and 16 hexagons, and additionally for 14, 17, and 23 hexagons after minimal expert tuning of the generated code. For the second autocorrelation inequality, the evolved and human-scaled program pushes the lower bound from 0.96102 to 0.96258. For spherical codes, the ImprovEvolve program lowers the best-known maximum cosine for the majority of 90 randomly chosen diverse state-of-the-art spherical codes, achieving relative improvements of up to 2.4%.

Abstract

Automated Algorithm Discovery

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