The Orthogonal Vectors Conjecture for Branching Programs and Formulas

TL;DR

This paper proves the Orthogonal Vectors Conjecture (OVC) in several computational models, establishing tight lower bounds for OV and related problems, and explores limitations of random restrictions in proving hardness.

Contribution

The paper establishes tight lower bounds for OV in branching programs and formulas, and introduces new input restriction techniques that challenge traditional random restriction methods.

Findings

OV has near-optimal complexity in branching programs and formulas.

Lower bounds match known quadratic bounds for explicit functions.

Random restrictions cannot prove OVC hardness in the average case.

Abstract

In the Orthogonal Vectors (OV) problem, we wish to determine if there is an orthogonal pair of vectors among $n$ Boolean vectors in $d$ dimensions. The OV Conjecture (OVC) posits that OV requires $n^{2-o(1)}$ time to solve, for all $d=\omega(\log n)$. Assuming the OVC, optimal time lower bounds have been proved for many prominent problems in $P$. We prove that OVC is true in several computational models of interest: * For all sufficiently large $n$ and $d$, OV for $n$ vectors in $\{0,1\}^d$ has branching program complexity $\tilde{\Theta}(n\cdot \min(n,2^d))$. In particular, the lower bounds match the upper bounds up to polylog factors. * OV has Boolean formula complexity $\tilde{\Theta}(n\cdot \min(n,2^d))$, over all complete bases of $O(1)$ fan-in. * OV requires $\tilde{\Theta}(n\cdot \min(n,2^d))$ wires, in formulas comprised of gates computing arbitrary symmetric functions of unbounded fan-in. Our lower bounds basically match the best known (quadratic) lower bounds for any explicit function in those models. Analogous lower bounds hold for many related problems shown to be hard under OVC, such as Batch Partial Match, Batch Subset Queries, and Batch Hamming Nearest Neighbors, all of which have very succinct reductions to OV. The proofs use a certain kind of input restriction that is different from typical random restrictions where variables are assigned independently. We give a sense in which independent random restrictions cannot be used to show hardness, in that OVC is false in the "average case" even for $AC^0$ formulas: * For every fixed $p \in (0,1)$ there is an $\epsilon_p > 0$ such that for every $n$ and $d$, OV instances where input bits are independently set to $1$ with probability $p$ (and $0$ otherwise) can be solved with $AC^0$ formulas of size $O(n^{2-\epsilon_p})$, on all but a $o_n(1)$ fraction of instances.