The Orthogonal Vectors Conjecture for Branching Programs and Formulas.

In the Orthogonal Vectors (OV) problem, we wish to determine if there is an orthogonal pair of vectors among $n$ Boolean vectors $d$ dimensions. 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 $P$. prove that OVC is true several computational models of interest: * For all sufficiently large $n$ and $d$, OV for $n$ vectors ${0,1}^d$ has branching program complexity $tilde{Theta}(ncdot min(n,2^d))$. In particular, the lower bounds match the upper bounds up to polylog factors. * OV has Boolean formula complexity $tilde{Theta}(ncdot min(n,2^d))$, over all complete bases of $O(1)$ fan-in. * OV requires $tilde{Theta}(ncdot min(n,2^d))$ wires, 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 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 which independent random restrictions cannot be used to show hardness, that OVC is false the average case even for $AC^0$ formulas: * For every fixed $p in (0,1)$ there is an $epsilon_p u003e 0$ 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.