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- poly (d) time for any constant 0 when d (n). 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 for 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 ( (n/t) ᵏ). In particular, when k is a constant, any disjunction of k -CNFs computing k -OV needs to use (nᵏ) CNFs. This matches the brute-force construction, and for each k 2, this is the first tight unconditional (nᵏ) lower bound against a class of circuits computing k -OV. Our results partially resolve a conjecture by Kane and Williams 26 (page 12, conjecture 10) about depth-3 AC⁰ circuits computing 2-OV by showing that the conjecture is true for circuits having bounded bottom fan-in. 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 for k -OV as well.