Box Facets and Cut Facets of Lifted Multicut Polytopes

Lucas Fabian Naumann, Jannik Irmai, Shengxian Zhao, Bjoern Andres
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:37365-37375, 2024.

Abstract

The lifted multicut problem has diverse applications in the field of computer vision. Exact algorithms based on linear programming require an understanding of lifted multicut polytopes. Despite recent progress, two fundamental questions about these polytopes have remained open: Which lower box inequalities define facets, and which cut inequalities define facets? In this article, we answer the first question by establishing conditions that are necessary, sufficient and efficiently decidable. Toward the second question, we show that deciding facet-definingness of cut inequalities is NP-hard. This completes the analysis of canonical facets of lifted multicut polytopes.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-naumann24a, title = {Box Facets and Cut Facets of Lifted Multicut Polytopes}, author = {Naumann, Lucas Fabian and Irmai, Jannik and Zhao, Shengxian and Andres, Bjoern}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {37365--37375}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/naumann24a/naumann24a.pdf}, url = {https://proceedings.mlr.press/v235/naumann24a.html}, abstract = {The lifted multicut problem has diverse applications in the field of computer vision. Exact algorithms based on linear programming require an understanding of lifted multicut polytopes. Despite recent progress, two fundamental questions about these polytopes have remained open: Which lower box inequalities define facets, and which cut inequalities define facets? In this article, we answer the first question by establishing conditions that are necessary, sufficient and efficiently decidable. Toward the second question, we show that deciding facet-definingness of cut inequalities is NP-hard. This completes the analysis of canonical facets of lifted multicut polytopes.} }
Endnote
%0 Conference Paper %T Box Facets and Cut Facets of Lifted Multicut Polytopes %A Lucas Fabian Naumann %A Jannik Irmai %A Shengxian Zhao %A Bjoern Andres %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-naumann24a %I PMLR %P 37365--37375 %U https://proceedings.mlr.press/v235/naumann24a.html %V 235 %X The lifted multicut problem has diverse applications in the field of computer vision. Exact algorithms based on linear programming require an understanding of lifted multicut polytopes. Despite recent progress, two fundamental questions about these polytopes have remained open: Which lower box inequalities define facets, and which cut inequalities define facets? In this article, we answer the first question by establishing conditions that are necessary, sufficient and efficiently decidable. Toward the second question, we show that deciding facet-definingness of cut inequalities is NP-hard. This completes the analysis of canonical facets of lifted multicut polytopes.
APA
Naumann, L.F., Irmai, J., Zhao, S. & Andres, B.. (2024). Box Facets and Cut Facets of Lifted Multicut Polytopes. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:37365-37375 Available from https://proceedings.mlr.press/v235/naumann24a.html.

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