keyisi/steiner-ratio

Steiner Ratio 0.8559 Certificates

This dataset contains the pre-generated binary partitions and certificate records accompanying the paper Towards Solving the Gilbert-Pollak Conjecture via Large Language Models, published at ICML 2026. They support the computer-assisted verification of the lower bound (\rho \ge 0.8559) for the Steiner ratio.

This is a research artifact for mathematical verification, not a conventional machine-learning dataset.

Related resources

• Source code and verifier: keyisi2006/Steiner-Ratio
• Verification instructions: certificate/README.md

Dataset contents

The dataset contains four cases, each with a certificate file and a child file.

Case	Variables (n)	Certificate records	Certificate size	Child-file size
d_regular/f_ge_d	5	1,888,501	173,742,092 bytes	30,215,760 bytes
d_regular/f_le_d	6	63,466,354	6,854,366,232 bytes	1,015,461,152 bytes
d_steiner/f_ge_d	7	5,402,950	669,965,800 bytes	86,446,176 bytes
d_steiner/f_le_d	8	259,435,950	36,321,033,000 bytes	4,150,973,152 bytes

The eight data files occupy approximately 49.3 GB in total.

Each case directory contains:

• certificate_rho=0.8559_f_ge_d.bin or certificate_rho=0.8559_f_le_d.bin: certified boxes and their selected (split_id, lemma_id) pairs.
• child_rho=0.8559_f_ge_d.bin or child_rho=0.8559_f_le_d.bin: parent-child relationships for verifying that the boxes form a complete partition.

Binary format

All values are stored in little-endian order without padding.

Each certificate record contains:

1. one int32 region ID;
2. n pairs of float64 lower and upper box bounds;
3. one int32 split ID;
4. one int32 lemma ID.

The record size is 12 + 16*n bytes.

Each child record contains two int32 child region IDs. Child records are stored in reverse region-ID order; use the verifier's --reversed option.

In the published d_regular/f_le_d certificate, boxes belonging to the other subcase use (split_id, lemma_id) = (0, 0) as a sentinel. The current generator uses (-1, -1). The verifier applies the subcase filter before reading the selected split, so the two encodings have the same meaning for these records.

Notes

Due to thread scheduling and platform-dependent floating-point behavior, running the same generation command may produce files that differ in record order, region IDs, or certificate choices, while preserving the same box partition.

License

The certificate and partition files in this dataset repository are released under the MIT License, Copyright (c) 2025-2026 Yisi Ke.

Citation

If you use this dataset, please cite the associated paper:

@inproceedings{ke2026towards,
  title   = {Towards Solving the Gilbert-Pollak Conjecture via Large Language Models},
  author  = {Ke, Yisi and Huang, Tianyu and Shu, Yankai and He, Di and Gai, Jingchu and Wang, Liwei},
  booktitle = {Proceedings of the 43rd International Conference on Machine Learning},
  year    = {2026},
  url     = {https://openreview.net/forum?id=Cqq3HGRIuM}
}