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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2509.19598 |
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| _version_ | 1866909810086117376 |
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| author | Compton, Spencer |
| author_facet | Compton, Spencer |
| contents | Given $m \ge 2$ discrete probability distributions over $n$ states each, the minimum-entropy coupling is the minimum-entropy joint distribution whose marginals are the same as the input distributions. Computing the minimum-entropy coupling is NP-hard, but there has been significant progress in designing approximation algorithms; prior to this work, the best known polynomial-time algorithms attain guarantees of the form $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + c$, where $c \approx 0.53$ for $m=2$, and $c \approx 1.22$ for general $m$ [CKQGK '23].
A main open question is whether this task is APX-hard, or whether there exists a polynomial-time approximation scheme (PTAS). In this work, we design an algorithm that produces a coupling with entropy $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + \varepsilon$ in running time $n^{O(\operatorname{poly}(1/\varepsilon) \cdot \operatorname{exp}(m) )}$: showing a PTAS exists for constant $m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19598 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Efficient $\varepsilon$-approximate minimum-entropy couplings Compton, Spencer Information Theory Data Structures and Algorithms Given $m \ge 2$ discrete probability distributions over $n$ states each, the minimum-entropy coupling is the minimum-entropy joint distribution whose marginals are the same as the input distributions. Computing the minimum-entropy coupling is NP-hard, but there has been significant progress in designing approximation algorithms; prior to this work, the best known polynomial-time algorithms attain guarantees of the form $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + c$, where $c \approx 0.53$ for $m=2$, and $c \approx 1.22$ for general $m$ [CKQGK '23]. A main open question is whether this task is APX-hard, or whether there exists a polynomial-time approximation scheme (PTAS). In this work, we design an algorithm that produces a coupling with entropy $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + \varepsilon$ in running time $n^{O(\operatorname{poly}(1/\varepsilon) \cdot \operatorname{exp}(m) )}$: showing a PTAS exists for constant $m$. |
| title | Efficient $\varepsilon$-approximate minimum-entropy couplings |
| topic | Information Theory Data Structures and Algorithms |
| url | https://arxiv.org/abs/2509.19598 |