heilbronn_problem_for_convex_regions

Heilbronn Problem for Convex Regions Example

This example demonstrates how to use the LoongFlow framework to solve a challenging computational geometry optimization problem. The goal is to evolve a Python algorithm that finds a specific configuration of points within a unit square.

Problem Description

The objective is to place $n$ points in a unit square $[0, 1] \times [0, 1]$ such that the minimum area of any triangle formed by three of these points is maximized. This is a variation of the classical Heilbronn triangle problem.

For the detailed mathematical definition and problem context, please refer to the official AlphaEvolve problem description: Heilbronn Problem Results

In this specific configuration ($n=13$), we aim to maximize the minimum triangle area.

Project Structure

• initial_program.py: The starting seed code. It contains a basic function signature find_best_placement and a simulated annealing implementation that needs to be evolved.
• eval_program.py: The evaluation logic. It executes the generated code in a secure/isolated manner, verifies geometric constraints (distinct points, non-collinear, inside unit square), and calculates the score based on the target area.
• task_config.yaml: The main configuration file defining the LLM prompt, evolution parameters (iterations, target score), and the agent components (Planner, Executor, Summarizer).

How to Run

To start the evolution process, you need to use the math_agent_agent.py entry point. Ensure your PYTHONPATH includes the project root so that python can find the agents and evolux modules.

1. Prerequisites

Ensure you are in the root directory of your local project (the directory containing agents/ and evolux/).

2. Execution Command

Run the following command to kick off the evolution. This command loads the base configuration and injects the initial code and evaluation logic from the respective files.

python agents/math_agent/math_agent_agent.py \
  --config agents/math_agent/examples/heilbronn_problem_for_convex_regions/task_config.yaml \
  --initial-file agents/math_agent/examples/heilbronn_problem_for_convex_regions/initial_program.py \
  --eval-file agents/math_agent/examples/heilbronn_problem_for_convex_regions/eval_program.py \
  --log-level INFO

Arguments Explanation:

• --config: Path to the YAML configuration file (task_config.yaml).
• --initial-file: Path to the Python file containing the seed code (initial_program.py). The content of this file will be injected into evolve.initial_code.
• --eval-file: Path to the Python file containing the evaluation logic (eval_program.py). The content will be injected into evolve.evaluator.evaluate_code.
• --log-level: Sets the logging verbosity (e.g., INFO, DEBUG).

3. Configuration Highlights

The task_config.yaml is pre-configured with the following strategies:

• Planner: evolve_planner (Handles the strategic direction of code modification).
• Executor: evolve_executor_fuse (A powerful executor that fuses multiple thought processes/candidates).
• Summarizer: evolve_summary (Summarizes the results of the execution for the next iteration).
• Target: The evolution aims for a target score of 1.0 (normalized against the benchmark area).

Evolution Process & Results

The system iterates through generations of code, attempting to maximize the minimum triangle area.

Final Result

The best solution found by LoongFlow achieved a minimum area of 0.030900663674639613, surpassing the previous SOTA benchmark of 0.0306.

Result Metrics:

• Optimized Minimum Area ($n=13$): 0.03090066

Troubleshooting

• TimeoutError: The eval_program.py enforces a strict timeout (default 3600s in config, though internal function calls have shorter timeouts). If the generated code enters an infinite loop, it will be terminated and marked as a failure.
• ModuleNotFoundError: Ensure your PYTHONPATH is set correctly. You may need to run export PYTHONPATH=$PYTHONPATH:. in the project root before running the command.