On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product
share
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 ℓ_2-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 {0,1}^d, there is an n^2 - Ω(1) time ( d/ n )^Ω(1)-multiplicative-approximating algorithm, and we show this is conditionally optimal, as such a (d/ n)^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 {0,1}^d, there is an n^2 - Ω(1) time Ω(d)-additive-approximating algorithm, and we show this is conditionally optimal, as such an o(d)-approximating algorithm would refute SETH. 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 Z^d for some d = 2^O(^* n), every exact algorithm requires n^2 - o(1) time. With the reduction from [Williams, SODA 2018], it follows that ℓ_2-Furthest Pair and Bichromatic ℓ_2-Closest Pair in 2^O(^* n) dimensions require n^2 - o(1) time.
READ FULL TEXT