The Planted Orthogonal Vectors Problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911152108208128 |
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| author | Kühnemann, David Polak, Adam Rosen, Alon |
| author_facet | Kühnemann, David Polak, Adam Rosen, Alon |
| contents | In the $k$-Orthogonal Vectors ($k$-OV) problem we are given $k$ sets, each containing $n$ binary vectors of dimension $d=n^{o(1)}$, and our goal is to pick one vector from each set so that at each coordinate at least one vector has a zero. It is a central problem in fine-grained complexity, conjectured to require $n^{k-o(1)}$ time in the worst case.
We propose a way to \emph{plant} a solution among vectors with i.i.d. $p$-biased entries, for appropriately chosen $p$, so that the planted solution is the unique one. Our conjecture is that the resulting $k$-OV instances still require time $n^{k-o(1)}$ to solve, \emph{on average}.
Our planted distribution has the property that any subset of strictly less than $k$ vectors has the \emph{same} marginal distribution as in the model distribution, consisting of i.i.d. $p$-biased random vectors. We use this property to give average-case search-to-decision reductions for $k$-OV. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Planted Orthogonal Vectors Problem Kühnemann, David Polak, Adam Rosen, Alon Computational Complexity Cryptography and Security Data Structures and Algorithms In the $k$-Orthogonal Vectors ($k$-OV) problem we are given $k$ sets, each containing $n$ binary vectors of dimension $d=n^{o(1)}$, and our goal is to pick one vector from each set so that at each coordinate at least one vector has a zero. It is a central problem in fine-grained complexity, conjectured to require $n^{k-o(1)}$ time in the worst case. We propose a way to \emph{plant} a solution among vectors with i.i.d. $p$-biased entries, for appropriately chosen $p$, so that the planted solution is the unique one. Our conjecture is that the resulting $k$-OV instances still require time $n^{k-o(1)}$ to solve, \emph{on average}. Our planted distribution has the property that any subset of strictly less than $k$ vectors has the \emph{same} marginal distribution as in the model distribution, consisting of i.i.d. $p$-biased random vectors. We use this property to give average-case search-to-decision reductions for $k$-OV. |
| title | The Planted Orthogonal Vectors Problem |
| topic | Computational Complexity Cryptography and Security Data Structures and Algorithms |
| url | https://arxiv.org/abs/2505.00206 |