The Planted Orthogonal Vectors Problem

• URL: http://arxiv.org/abs/2505.00206v1
• Date: Wed, 30 Apr 2025 22:13:11 GMT
• Title: The Planted Orthogonal Vectors Problem
• Authors: David Kühnemann, Adam Polak, Alon Rosen,
• Abstract summary: We find a way to emphplant a solution among vectors with i.i.d. $p$-biased entries, for appropriately chosen $p$, so that the planted solution is the unique one.<n>Our conjecture is that the resulting $k$-OV instances still require time $nk-o(1)$ to solve.
• Score: 4.017450863157041
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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