Minimal Unimodal Decomposition is NP-Hard on Graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909829599068160 |
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| author | K, Mishal Assif P Baryshnikov, Yuliy |
| author_facet | K, Mishal Assif P Baryshnikov, Yuliy |
| contents | A function on a topological space is called unimodal if all of its super-level sets are contractible. A minimal unimodal decomposition of a function $f$ is the smallest number of unimodal functions that sum up to $f$. The problem of decomposing a given density function into its minimal unimodal components is fundamental in topological statistics. We show that finding a minimal unimodal decomposition of an edge-linear function on a graph is NP-hard. Given any $k \geq 2$, we establish the NP-hardness of finding a unimodal decomposition consisting of $k$ unimodal functions. We also extend the NP-hardness result to related variants of the problem, including restriction to planar graphs, inapproximability results, and generalizations to higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05944 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal Unimodal Decomposition is NP-Hard on Graphs K, Mishal Assif P Baryshnikov, Yuliy Algebraic Topology Computational Geometry Statistics Theory 55N31, 68Q17, 65C60, 68R10 A function on a topological space is called unimodal if all of its super-level sets are contractible. A minimal unimodal decomposition of a function $f$ is the smallest number of unimodal functions that sum up to $f$. The problem of decomposing a given density function into its minimal unimodal components is fundamental in topological statistics. We show that finding a minimal unimodal decomposition of an edge-linear function on a graph is NP-hard. Given any $k \geq 2$, we establish the NP-hardness of finding a unimodal decomposition consisting of $k$ unimodal functions. We also extend the NP-hardness result to related variants of the problem, including restriction to planar graphs, inapproximability results, and generalizations to higher dimensions. |
| title | Minimal Unimodal Decomposition is NP-Hard on Graphs |
| topic | Algebraic Topology Computational Geometry Statistics Theory 55N31, 68Q17, 65C60, 68R10 |
| url | https://arxiv.org/abs/2510.05944 |