SAT-Based Search for Minwise Independent Families
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929633309491200 |
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| author | Iurlano, Enrico Raidl, Günther R. |
| author_facet | Iurlano, Enrico Raidl, Günther R. |
| contents | Proposed for rapid document similarity estimation in web search engines, the celebrated property of minwise independence imposes highly symmetric constraints on a family $\mathcal{F}$ of permutations of $\{1,\ldots, n\}$: The property is fulfilled by $\mathcal{F}$ if for each $j\in \{1,\ldots,n\}$, any cardinality-$j$ subset $X\subseteq \{1,\ldots,n\}$, and any fixed element $x^\ast\in X$, it occurs with probability $1/j$ that a randomly drawn permutation $π$ from $\mathcal{F}$ satisfies $π(x^\ast)=\min \{π(x) : x\in X\}$. The central interest is to find a family with fewest possible members meeting the stated constraints. We provide a framework that, firstly, is realized as a pure SAT model and, secondly, generalizes a heuristic of Mathon and van Trung to the search of these families. Originally, the latter enforces an underlying group-theoretic decomposition to achieve a significant speed-up for the computer-aided search of structures which can be identified with so-called rankwise independent families. We observe that this approach is suitable to find provenly optimal new representatives of minwise independent families while yielding a decisive speed-up, too. As the problem has a naive search space of size at least $(n!)^n$, we also carefully address symmetry breaking. Finally, we add a bijective proof for a problem encountered by Bargachev when deriving a lower bound on the number of members in a minimal rankwise independent family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | SAT-Based Search for Minwise Independent Families Iurlano, Enrico Raidl, Günther R. Discrete Mathematics 90C27, 68R05 Proposed for rapid document similarity estimation in web search engines, the celebrated property of minwise independence imposes highly symmetric constraints on a family $\mathcal{F}$ of permutations of $\{1,\ldots, n\}$: The property is fulfilled by $\mathcal{F}$ if for each $j\in \{1,\ldots,n\}$, any cardinality-$j$ subset $X\subseteq \{1,\ldots,n\}$, and any fixed element $x^\ast\in X$, it occurs with probability $1/j$ that a randomly drawn permutation $π$ from $\mathcal{F}$ satisfies $π(x^\ast)=\min \{π(x) : x\in X\}$. The central interest is to find a family with fewest possible members meeting the stated constraints. We provide a framework that, firstly, is realized as a pure SAT model and, secondly, generalizes a heuristic of Mathon and van Trung to the search of these families. Originally, the latter enforces an underlying group-theoretic decomposition to achieve a significant speed-up for the computer-aided search of structures which can be identified with so-called rankwise independent families. We observe that this approach is suitable to find provenly optimal new representatives of minwise independent families while yielding a decisive speed-up, too. As the problem has a naive search space of size at least $(n!)^n$, we also carefully address symmetry breaking. Finally, we add a bijective proof for a problem encountered by Bargachev when deriving a lower bound on the number of members in a minimal rankwise independent family. |
| title | SAT-Based Search for Minwise Independent Families |
| topic | Discrete Mathematics 90C27, 68R05 |
| url | https://arxiv.org/abs/2412.11811 |