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

Abstract

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 e2-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.• Characterization of Multiplicative Approximation. 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 n2−ω(1) time (d/logn)ω(1) - multiplicative-approximating algorithm, and we show this is conditionally optimal, as such a (d/logn)o(1)-approximating algorithm would refute SETH. Similar characterization is also achieved for additive approximation for Max-IP.• 2O(log* n)-dimensional Hardness for Exact Max-IP Over The Integers. Second, 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 Zd for some d = 2O(log* n), every exact algorithm requires n2−o(1) time. With the reduction from [Williams, SODA 2018], it follows that e2-Furthest Pair and Bichromatic e2-Closest Pair in 2O(log* n) dimensions require n2−o(1) time.• Connection with NP · UPP Communication Protocols. Last, We establish a connection between conditional lower bounds for exact Max-IP with integer entries and NP · UPP communication protocols for Set-Disjointness, parallel to the connection between conditional lower bounds for approximating Max-IP and MA communication protocols for Set-Disjointness.The lower bound in our first result is a direct corollary of the new MA protocol for Set-Disjointness introduced in [Rubinstein, STOC 2018], and our algorithms utilize the polynomial method and simple random sampling. Our second result follows from a new dimensionality self reduction from the Orthogonal Vectors problem for n vectors from {0, 1}d to n vectors from Ze where e = 2O(log* d), dramatically improving the previous reduction in [Williams, SODA 2018]. The key technical ingredient is a recursive application of Chinese Remainder Theorem.As a side product, we obtain an MA communication protocol for Set-Disjointness with complexity [EQUATION], slightly improving the [EQUATION] log n) bound [Aaronson and Wigderson, TOCT 2009], and approaching the [EQUATION] lower bound [Klauck, CCC 2003].Moreover, we show that (under SETH) one can apply the [EQUATION] BQP communication protocol for Set-Disjointness to prove near-optimal hardness for approximation to Max-IP with vectors in {− 1, 1}d. This answers a question from [Abboud et al., FOCS 2017] in the affirmative.