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

Abstract: We present Discovery Loop, a lightweight system that uses a 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.

• The paper demonstrates Discovery Loop, an LLM-guided program evolution method that improved packing records at Packomania.
• Discovery Loop operates with a single active champion and no explicit population.
• For a consumer-grade setup and a budget of only $28, the method produced independently verified, substantial improvements in circle packing for 10 target values.

Problem setting and research objective

“LLM-Guided Program Evolution for Circle Packing: Breaking 10 Packomania Records for $28” presents Discovery Loop, a minimal framework for LLM-guided algorithm evolution applied to variable-radius circle packing (2609.05093). The paper investigates whether the central mechanism of systems such as AlphaEvolve and FunSearch can produce meaningful optimization advances without distributed infrastructure, population-based evolutionary control, or substantial computational expenditure.

The benchmark is the Packomania csqv problem: place $N$$circles with variable radii inside the unit square while preventing overlap and maximizing the sum of radii. For circle$$i$, with center$(x_i,y_i)$and radius$r_i$$, feasibility requires containment within the square and pairwise separation by at least$$r_i+r_j$. The objective is$\sum_i r_i$$. This formulation is highly nonconvex, and effective solvers must combine continuous optimization of coordinates and radii with mechanisms capable of changing the combinatorial structure of contacts and local arrangements.</p> <p>The experimental targets comprised twelve values of$$N$, including ten instances from$N=101$to$114$and two smaller instances,$N=26$and$i$0. The principal claim is that Discovery Loop improved the Packomania records for ten of these targets, with gains between $i$1 and $i$227.72. The resulting solutions were independently verified and accepted by Packomania.

Discovery Loop architecture

Discovery Loop contains a loop controller, a problem-specific plugin, and an initial seed solver. The controller maintains a current champion solver, per-target best results, a scoreboard, and a history of previously attempted ideas. At each iteration, the LLM receives the complete source code of the current champion, the problem formulation, target-wise scores, and the outcomes of recent attempts. It then produces a complete replacement solver together with an informal description of its proposed algorithmic change.

This full-replacement strategy is central to the design. Rather than requiring the model to produce a syntactically valid patch against an evolving codebase, the system asks for a self-contained solver that can alter the optimization architecture. The paper argues that this permits transitions between qualitatively different strategies, such as replacing random initialization with lattice-based initialization or introducing a specialized KKT polish stage.

Candidate solvers are evaluated independently on all targets, using six parallel workers and a timeout of 120 seconds per target. A candidate becomes the new champion only if its aggregate score exceeds that of the current champion. However, the system separately retains the best result observed for every target. This distinction prevents a solver that is globally inferior from eliminating a specialized result on a particular instance. It also acknowledges that a single solver may not dominate across all target sizes.

The initial solver combines multi-start penalty optimization using L-BFGS-B with linear-programming optimization of the radii. Subsequent proposals modify initialization, basin exploration, local polishing, and structural perturbation. The evolutionary state is therefore small: there is no explicit population, island archive, or multi-parent recombination at the meta-level. The system maintains one active champion and a textual history of attempts.

Independent verification and evaluation integrity

The verifier is deliberately separated from the solver. It checks square containment, pairwise non-overlap, and the independently recomputed objective value. It also applies a strict feasibility shrink intended to remove numerical artifacts arising from floating-point optimization. The paper describes the checker as operating with zero tolerance, which is important for a benchmark where small infeasibilities can otherwise produce apparently superior objective values.

This architecture addresses a central vulnerability of LLM-generated optimization code: the candidate program must not be able to exploit an evaluator that shares implementation assumptions or numerical shortcuts with the solver. Independent verification makes the reported records substantially more credible. It does not, however, establish global optimality. The results demonstrate feasible improvements over the Packomania records, which are best-known solutions rather than necessarily proven optima.

Record-breaking results

The strongest empirical result is the improvement on ten large instances. The gains are concentrated in the range $i$3--$i$4 and are consistently substantial relative to the prior records.

$i$5	Prior record	Discovery Loop	Improvement
101	5.163845	5.289154	2.43%
102	5.055187	5.318238	5.20%
103	5.085509	5.345481	5.11%
105	5.125967	5.401298	5.37%
106	5.151736	5.429079	5.38%
107	5.180124	5.453952	5.29%
108	5.205806	5.481819	5.30%
109	5.231096	5.507926	5.29%
111	5.278427	5.554909	5.24%
114	5.336683	5.624188	5.39%

Aggregated over these ten targets, the prior-record sum increased from $i$6 to $i$7, a $i$8 improvement. The two remaining targets, $i$9 and $(x_i,y_i)$0, matched or approached their prior records within $(x_i,y_i)$1 but did not exceed them.

A significant qualification is that all ten record improvements were already obtained by the seed solver at iteration 0. The LLM-guided iterations therefore improved the aggregate objective after the system had already surpassed the listed records, but they were not responsible for initially breaking those ten records. This distinction matters for interpreting the headline claim: the experiment demonstrates the value of the complete pipeline, including the manually designed seed solver, while the incremental contribution of LLM evolution is reflected in later aggregate improvements rather than in the first record-breaking solutions.

Algorithmic proposals generated by the LLM

The candidate sequence contains several technically plausible modifications. The first accepted proposal introduced basin hopping around incumbent solutions, followed by SLSQP polishing on the resulting contact structure. This replaced purely cold-start exploration with exploitation of promising configurations.

A later candidate introduced hexagonal-lattice templates sized for the target number of circles. This proposal is particularly relevant to the geometry of dense packing because structured initial configurations can access regions of the search space unlikely to be reached by independent random initialization. The paper reports this as an architectural change rather than a parameter adjustment.

Other accepted proposals included island-model parallel basin hopping with elite migration, affine lattice template banks, a specialized KKT-Newton polish, lattice-aware slip moves, and defect-migration operators. Defect migration removes weak circles, allows the remaining configuration to re-equilibrate, and reinserts circles into newly available holes. Such an operator changes the contact graph and therefore addresses a limitation of local continuous optimization, which may be unable to cross structural barriers between distinct packing arrangements.

The paper characterizes these proposals as “genuine algorithmic innovations.” The claim is plausible in the narrow sense that the candidates introduce new search operators and solver components rather than merely changing scalar hyperparameters. Nevertheless, the experiment does not provide ablations isolating the causal contribution of each component. It is therefore not possible to determine from the reported results whether the improvements arise from individual innovations, interactions among them, longer computation, or stochastic variation in the underlying solver.

Iteration dynamics and cost efficiency

The system executed fifteen iterations before reaching the $(x_i,y_i)$259.39$(x_i,y_i)$359.98$(x_i,y_i)$44.96$(x_i,y_i)$50.57$(x_i,y_i)$622.76$(x_i,y_i)$70.02$$of improvement.</p> <p>The corresponding cost per unit of aggregate improvement increased from approximately$$(x_i,y_i)$8 in the early phase to approximately $(x_i,y_i)$9 in the later phase, a reported $r_i$0-fold deterioration. This result is one of the paper’s clearest operational findings. On a fixed set of targets, continuing LLM-guided search after the initial improvements can be economically irrational even when occasional candidates are accepted.

The paper implements a plateau detector using a four-iteration window. It stops when recent iterations contain no champion, no usable code, or less than $r_i$1 aggregate improvement. Backtesting against the observed run suggests that stopping after iteration 9 would have reduced expenditure from $r_i$213.95, a nominal saving of approximately $r_i$30.006$r_i$40.01\%$ of the final value.

This result should be interpreted as retrospective rather than prospective evidence. The threshold and window were evaluated on the same trajectory used to motivate them, so the reported savings do not establish generalization to other problems, models, or random seeds. They nonetheless provide a concrete stopping heuristic for expensive program-search loops.

Comparison with larger program-evolution systems

Discovery Loop occupies a substantially smaller design point than AlphaEvolve and FunSearch. It uses one LLM call per iteration, local parallel evaluation, a single active champion, and full-program replacement. AlphaEvolve and FunSearch employ more elaborate evaluation and search infrastructure, including distributed execution and broader evolutionary control. The comparison therefore concerns accessibility rather than matched performance.

The reported cost of $27.72 and execution on a consumer machine are important engineering facts. They show that some forms of LLM-guided algorithm search can be conducted by an individual researcher without access to a cluster. However, the comparison does not establish that Discovery Loop achieves comparable algorithmic quality to those larger systems. The experiment uses one benchmark family, one model, one seed solver, and one trajectory. Cost and infrastructure requirements are therefore better supported than general claims about relative discovery capability.

The full-program replacement interface also creates a trade-off. It makes major architectural changes easy to express, but it discards potentially useful modularity and inheritance between candidates. A population-based system could preserve diverse solvers, recombine successful components, and reduce dependence on a single champion trajectory. Discovery Loop does not test whether its simplicity is advantageous beyond this specific setting.

Limitations and open questions

The evidence is limited to a single optimization domain whose formulation, evaluation procedure, and computational cost are unusually favorable to LLM-guided search. Circle packing has a compact mathematical description, candidate evaluation is parallelizable, and the solver can use standard numerical optimization libraries. The paper does not demonstrate comparable performance on problems with expensive evaluation, complex interfaces, discrete constraints, or difficult code-generation requirements.

Model dependence is also substantial. The experiment uses Claude Fable 5.1 through the Claude CLI, and the paper reports a separate MIPLIB experiment with a 75% code-generation failure rate. This observation directly qualifies the accessibility claim: low monetary cost does not imply reliable algorithm discovery when the target problem requires more complex or precise generated programs.

The experimental design lacks repeated runs, alternative LLMs, random-seed analysis, and controlled ablations. Consequently, the robustness of the improvements and the contribution of individual operators remain unresolved. The fact that the seed solver produced all ten record-breaking results further complicates attribution. A stronger evaluation would compare the seed and evolved solvers over repeated stochastic trials, report per-target distributions rather than only best outcomes, and evaluate candidates under matched computational budgets.

The plateau analysis also leaves an open methodological question. Its retrospective backtest suggests large savings, but the stopping rule was not validated online across independent runs. It remains unknown whether the same threshold would prematurely terminate searches whose improvements arrive intermittently or whose useful innovations require several consecutive unsuccessful proposals.

Finally, the system’s per-target best tracking produces a collection of specialized solutions rather than necessarily one universally superior solver. This is appropriate for reporting benchmark records, but it raises a distinction between benchmark improvement and general algorithm discovery. The paper leaves open whether the evolved components transfer to unseen values of $r_i$5, different container geometries, or related packing objectives.

Conclusion

Discovery Loop demonstrates that a compact LLM-driven replacement loop can generate useful optimization algorithms on a consumer machine at low monetary cost. In the reported circle-packing experiment, the pipeline produced independently verified improvements on ten Packomania instances, while later iterations yielded only marginal gains at sharply increasing cost. The results support the practical value of structured feedback, independent verification, and adaptive stopping, but they do not yet establish broad transferability or isolate the causal contribution of LLM evolution from the seed solver and stochastic numerical search.