Computing $p$-presentation distances is hard
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908395574919168 |
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| author | Bjerkevik, Håvard Bakke Botnan, Magnus Bakke |
| author_facet | Bjerkevik, Håvard Bakke Botnan, Magnus Bakke |
| contents | Recently, $p$-presentation distances for $p\in [1,\infty]$ were introduced for merge trees and multiparameter persistence modules as more sensitive variations of the respective interleaving distances ($p=\infty)$. It is well-known that computing the interleaving distance is NP-hard in both cases. We extend this result by showing that computing the $p$-presentation distance is NP-hard for all $p\in [1,\infty)$ for both merge trees and $t$-parameter persistence modules for any $t\geq 2$. Though the details differ, both proofs follow the same novel strategy, suggesting that our approach can be adapted to proving the NP-hardness of other distances based on sums or $p$-norms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07200 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing $p$-presentation distances is hard Bjerkevik, Håvard Bakke Botnan, Magnus Bakke Computational Geometry Computational Complexity Representation Theory Recently, $p$-presentation distances for $p\in [1,\infty]$ were introduced for merge trees and multiparameter persistence modules as more sensitive variations of the respective interleaving distances ($p=\infty)$. It is well-known that computing the interleaving distance is NP-hard in both cases. We extend this result by showing that computing the $p$-presentation distance is NP-hard for all $p\in [1,\infty)$ for both merge trees and $t$-parameter persistence modules for any $t\geq 2$. Though the details differ, both proofs follow the same novel strategy, suggesting that our approach can be adapted to proving the NP-hardness of other distances based on sums or $p$-norms. |
| title | Computing $p$-presentation distances is hard |
| topic | Computational Geometry Computational Complexity Representation Theory |
| url | https://arxiv.org/abs/2403.07200 |