Hybrid Simulated Annealing with Cosine Cooling and Levy Flights for Circle Packing

Keywords: Circle packing, Algorithm, Optimization, Simulated annealing

Abstract: The circle packing problem—arranging non-overlapping circles within a bounded domain to maximize a chosen metric—arises in computational geometry, material science, and visual design. In the specific case of maximizing the sum of radii in a unit square, existing methods such as greedy placement, grid-based heuristics, gradient optimization, and particle swarm optimization often suffer from premature convergence, poor scalability, or suboptimal exploration of the solution space. We present a novel hybrid algorithm that combines latin hypercube sampling with a modified simulated annealing procedure incorporating cosine-annealing temperature decay, occasional L\'evy-flight-inspired perturbations to escape local optima, and a dynamically shrinking local search radius. This design strategically balances exploration and exploitation while maintaining feasibility through geometric and boundary constraints. $\textbf{Our algorithm generates a new world record score of 2.6359372 on 26 circles}$, exceeding the best-known hand-crafted algorithms and recent Google AlphaEvolve solution ($2.634$ and $2.6358627$, respectively). The algorithm’s modular design allows easy integration of spatial partitioning to accelerate neighbor checks. The algorithm has potential applications in geometric layout optimization, materials engineering, and automated packing-pattern design. The source code is publicly available at: https://anonymous.4open.science/r/AI-AlgorithmResearcher-161C.

Submission Number: 42