Depth-3 Circuit Lower Bounds for k-OV

The 2-Orthogonal Vectors (2-OV) problem is the following: given two tuples A and B of n Boolean vectors, each of dimension d , decide if there exist vectors u ∈ A , and v ∈ B , such that u and v are orthogonal. This problem, and its generalization k -OV defined analogously for k tuples, are central problems in the area of fine-grained complexity. One of the major conjectures in fine-grained complexity is that k -OV cannot be solved by a randomised algorithm in n k − ϵ poly ( d ) time for any constant ϵ > 0. In this paper, we are interested in unconditional lower bounds against k -OV, but for weaker models of computation than the general Turing Machine. In particular, we are interested in circuit lower bounds to computing k -OV by Boolean circuit families of depth 3 of the form OR-AND-OR, or equivalently, a disjunction of CNFs . We show that for all k ≤ d , any disjunction of t -CNFs computing k -OV requires size Ω(( n/t ) k ). In particular, when k is a constant, any disjunction of k -CNFs computing k -OV needs to use Ω( n k ) CNFs. This matches the brute-force construction, and for each fixed k > 2, this is the first unconditional Ω( n k ) lower bound against k -OV for a computation model that can compute it in size O ( n k ). Our results partially resolve a conjecture by Kane and Williams [17] (page 12, conjecture 10) about depth-3 AC 0 circuits computing 2-OV. As a secondary result, we show an exponential lower bound on the size of AND ◦ OR ◦ AND circuits computing 2-OV when d is very large. Since 2-OV reduces to k -OV by projections trivially, this lower bound works against k -OV as well.