The orthogonal vectors conjecture for branching programs and formulas

Abstract

© Daniel M. Kane and R. Ryan Williams. 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 n2−o(1) time to solve, for all d = ω(log n). Assuming the OVC, optimal time lower bounds have been proved for many prominent problems in P, such as Edit Distance, Frechet Distance, Longest Common Subsequence, and approximating the diameter of a graph. 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 Θ(n · min(n, 2d)). In particular, the lower and upper bounds match up to polylog factors. ▬ OV has Boolean formula complexity Θ(n · min(n, 2d)), over all complete bases of O(1) fan-in. ▬ OV requires Θ(n · min(n, 2d)) 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 AC0 formulas: For all p ∈ (0, 1) there is a δp > 0 such that for every n and d, OV instances with input bits independently set to 1 with probability p (and 0 otherwise) can be solved with AC0 formulas of O(n2−δp ) size, on all but a on(1) fraction of instances. Moreover, limp→1 δp = 1.