On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product

In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets A and B of vectors, and the goal is to find a ∈ A and b ∈ B maximizing inner product a · b. Max-IP is very basic and serves as the base problem in the recent breakthrough of [Abboud et al., FOCS 2017] on hardness of approximation for polynomial-time problems. It is also used (implicitly) in the argument for hardness of exact ℓ2-Furthest Pair (and other important problems in computational geometry) in poly-log-log dimensions in [Williams, SODA 2018]. We have three main results regarding this problem. First, we study the best multiplicative approximation ratio for Boolean Max-IP in sub-quadratic time. We show that, for Max-IP with two sets of n vectors from 0,1d, there is an n2 - Omega(1) time left( d/log n right)Omega(1)-multiplicative-approximating algorithm, and we show this is conditionally optimal, as such a left(d/log nright)o(1)-approximating algorithm would refute SETH. Second, we achieve a similar characterization for the best additive approximation error to Boolean Max-IP. We show that, for Max-IP with two sets of n vectors from 0,1d, there is an n2 - Omega(1) time Omega(d)-additive-approximating algorithm, and this is conditionally optimal, as such an o(d)-approximating algorithm would refute SETH [Rubinstein, STOC 2018]. Last, we revisit the hardness of solving Max-IP exactly for vectors with integer entries. We show that, under SETH, for Max-IP with sets of n vectors from mathbbZd for some d = 2^O(log* n), every exact algorithm requires n2 - o(1) time. With the reduction from [Williams, SODA 2018], it follows that ℓ2-Furthest Pair and Bichromatic ℓ2-Closest Pair in 2^O(log* n) dimensions require n2 - o(1) time.