Even with AI, Bijection Discovery is Still Hard: The Opportunities and Challenges of OpenEvolve for Novel Bijection Construction

Helen Jenne, Davis Brown, Jesse He, Max Vargas, Henry Kvinge
Proceedings of the 2nd Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2026), PMLR 334(2):108-122, 2026.

Abstract

Evolutionary program synthesis systems such as AlphaEvolve, OpenEvolve, and ShinkaEvolve offer a new approach to AI-assisted mathematical discovery. These systems use large language models (LLMs) to generate candidate solutions to a problem as human readable code, which is then evolved to improve beyond single-shot outputs. While existing mathematical applications have mostly focused on problems of establishing bounds (e.g., sphere packing), the program synthesis approach is well suited to any problem where the solution takes the form of an explicit construction. With this in mind, in this paper we explore the use of OpenEvolve for combinatorial bijection discovery. We describe the results of applying OpenEvolve to three bijection construction problems involving Dyck paths, two of which are known and one of which is open. We find that while systems like OpenEvolve show promise as a valuable tool for combinatorialists, the problem of finding novel, research-level bijections remains a challenging task for current frontier systems, reinforcing the need for human mathematicians in the loop.

Cite this Paper

BibTeX
@InProceedings{pmlr-v334-jenne26a, title = {Even with AI, Bijection Discovery is Still Hard: The Opportunities and Challenges of OpenEvolve for Novel Bijection Construction}, author = {Jenne, Helen and Brown, Davis and He, Jesse and Vargas, Max and Kvinge, Henry}, booktitle = {Proceedings of the 2nd Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2026)}, pages = {108--122}, year = {2026}, editor = {Berman, Eddie and Bernárdez, Guillermo and Chen, Samantha and Cloninger, Alex and Doster, Timothy and Emerson, Tegan and Grigsby, J. Elisenda and Kvinge, Henry and Lawrence, Hannah and Marrinan, Tim and Myers, Audun and Papillon, Mathilde and Tahmasebi, Behrooz and Telyatnikov, Lev and Walters, Robin and Weber, Melanie and Xie, YuQing and Yeats, Eric}, volume = {334}, number = {2}, series = {Proceedings of Machine Learning Research}, month = {18--20 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v334/main/assets/jenne26a/jenne26a.pdf}, url = {https://proceedings.mlr.press/v334/jenne26a.html}, abstract = {Evolutionary program synthesis systems such as AlphaEvolve, OpenEvolve, and ShinkaEvolve offer a new approach to AI-assisted mathematical discovery. These systems use large language models (LLMs) to generate candidate solutions to a problem as human readable code, which is then evolved to improve beyond single-shot outputs. While existing mathematical applications have mostly focused on problems of establishing bounds (e.g., sphere packing), the program synthesis approach is well suited to any problem where the solution takes the form of an explicit construction. With this in mind, in this paper we explore the use of OpenEvolve for combinatorial bijection discovery. We describe the results of applying OpenEvolve to three bijection construction problems involving Dyck paths, two of which are known and one of which is open. We find that while systems like OpenEvolve show promise as a valuable tool for combinatorialists, the problem of finding novel, research-level bijections remains a challenging task for current frontier systems, reinforcing the need for human mathematicians in the loop.} }
Endnote
%0 Conference Paper %T Even with AI, Bijection Discovery is Still Hard: The Opportunities and Challenges of OpenEvolve for Novel Bijection Construction %A Helen Jenne %A Davis Brown %A Jesse He %A Max Vargas %A Henry Kvinge %B Proceedings of the 2nd Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2026) %C Proceedings of Machine Learning Research %D 2026 %E Eddie Berman %E Guillermo Bernárdez %E Samantha Chen %E Alex Cloninger %E Timothy Doster %E Tegan Emerson %E J. Elisenda Grigsby %E Henry Kvinge %E Hannah Lawrence %E Tim Marrinan %E Audun Myers %E Mathilde Papillon %E Behrooz Tahmasebi %E Lev Telyatnikov %E Robin Walters %E Melanie Weber %E YuQing Xie %E Eric Yeats %F pmlr-v334-jenne26a %I PMLR %P 108--122 %U https://proceedings.mlr.press/v334/jenne26a.html %V 334 %N 2 %X Evolutionary program synthesis systems such as AlphaEvolve, OpenEvolve, and ShinkaEvolve offer a new approach to AI-assisted mathematical discovery. These systems use large language models (LLMs) to generate candidate solutions to a problem as human readable code, which is then evolved to improve beyond single-shot outputs. While existing mathematical applications have mostly focused on problems of establishing bounds (e.g., sphere packing), the program synthesis approach is well suited to any problem where the solution takes the form of an explicit construction. With this in mind, in this paper we explore the use of OpenEvolve for combinatorial bijection discovery. We describe the results of applying OpenEvolve to three bijection construction problems involving Dyck paths, two of which are known and one of which is open. We find that while systems like OpenEvolve show promise as a valuable tool for combinatorialists, the problem of finding novel, research-level bijections remains a challenging task for current frontier systems, reinforcing the need for human mathematicians in the loop.
APA
Jenne, H., Brown, D., He, J., Vargas, M. & Kvinge, H.. (2026). Even with AI, Bijection Discovery is Still Hard: The Opportunities and Challenges of OpenEvolve for Novel Bijection Construction. Proceedings of the 2nd Conference on Topology, Algebra, and Geometry in Data Science(TAG-DS 2026), in Proceedings of Machine Learning Research 334(2):108-122 Available from https://proceedings.mlr.press/v334/jenne26a.html.

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