Formula Size-Depth Tradeoffs for Iterated Sub-Permutation Matrix Multiplication
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917702921093120 |
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| author | Rossman, Benjamin |
| author_facet | Rossman, Benjamin |
| contents | We study the formula complexity of Iterated Sub-Permutation Matrix Multiplication, the logspace-complete problem of computing the product of $k$ $n$-by-$n$ Boolean matrices with at most a single $1$ in each row and column. For all $d \le \log k$, this problem is solvable by $n^{O(dk^{1/d})}$ size monotone formulas of two distinct types: (unbounded fan-in) $AC^0$ formulas of depth $d+1$ and (semi-unbounded fan-in) $SAC^0$ formulas of $\bigwedge$-depth $d$ and $\bigwedge$-fan-in $k^{1/d}$. The results of this paper give matching $n^{Ω(dk^{1/d})}$ lower bounds for monotone $AC^0$ and $SAC^0$ formulas for all $k \le \log\log n$, as well as slightly weaker $n^{Ω(dk^{1/2d})}$ lower bounds for non-monotone $AC^0$ and $SAC^0$ formulas. These size-depth tradeoffs converge at $d = \log k$ to tight $n^{Ω(\log k)}$ lower bounds for both unbounded-depth monotone formulas [Ros15] and bounded-depth non-monotone formulas [Ros18]. Our non-monotone lower bounds extend to the more restricted Iterated Permutation Matrix Multiplication problem, improving the previous $n^{k^{1/\exp(O(d))}}$ tradeoff for this problem [BIP98]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16015 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Formula Size-Depth Tradeoffs for Iterated Sub-Permutation Matrix Multiplication Rossman, Benjamin Computational Complexity Combinatorics 68Q17 F.1.3; F.2.3 We study the formula complexity of Iterated Sub-Permutation Matrix Multiplication, the logspace-complete problem of computing the product of $k$ $n$-by-$n$ Boolean matrices with at most a single $1$ in each row and column. For all $d \le \log k$, this problem is solvable by $n^{O(dk^{1/d})}$ size monotone formulas of two distinct types: (unbounded fan-in) $AC^0$ formulas of depth $d+1$ and (semi-unbounded fan-in) $SAC^0$ formulas of $\bigwedge$-depth $d$ and $\bigwedge$-fan-in $k^{1/d}$. The results of this paper give matching $n^{Ω(dk^{1/d})}$ lower bounds for monotone $AC^0$ and $SAC^0$ formulas for all $k \le \log\log n$, as well as slightly weaker $n^{Ω(dk^{1/2d})}$ lower bounds for non-monotone $AC^0$ and $SAC^0$ formulas. These size-depth tradeoffs converge at $d = \log k$ to tight $n^{Ω(\log k)}$ lower bounds for both unbounded-depth monotone formulas [Ros15] and bounded-depth non-monotone formulas [Ros18]. Our non-monotone lower bounds extend to the more restricted Iterated Permutation Matrix Multiplication problem, improving the previous $n^{k^{1/\exp(O(d))}}$ tradeoff for this problem [BIP98]. |
| title | Formula Size-Depth Tradeoffs for Iterated Sub-Permutation Matrix Multiplication |
| topic | Computational Complexity Combinatorics 68Q17 F.1.3; F.2.3 |
| url | https://arxiv.org/abs/2406.16015 |