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 $a \in A$ and $b \in B$ maximizing inner product $a \cdot b$. 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 $\ell_2$-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.

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 $n^{2 - \Omega(1)}$ time $\left( d/\log n \right)^{\Omega(1)}$-multiplicative-approximating algorithm, and we show this is conditionally optimal, as such a $\left(d/\log n\right)^{o(1)}$-approximating algorithm would refute SETH.

Characterization of Additive Approximation. 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 $n^{2 - \Omega(1)}$ time $\Omega(d)$-additive-approximating algorithm, and this is conditionally optimal, as such an $o(d)$-approximating algorithm would refute SETH~[Rubinstein, STOC 2018].

$2^{O(\log^{*} n)}$-dimensional Hardness for Exact Max-IP Over The Integers. 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 $\mathbb{Z}^{d}$ for some $d = 2^{O(\log^{*} n)}$, every exact algorithm requires $n^{2 - o(1)}$ time. With the reduction from [Williams, SODA 2018], it follows that $\ell_2$-Furthest Pair and Bichromatic $\ell_2$-Closest Pair in $2^{O(\log^{*} n)}$ dimensions require $n^{2 - o(1)}$ time.

The lower bound in our first result is a direct corollary of the new MA protocol for Set-Disjointness introduced in [Rubinstein, STOC 2018]. Our algorithms utilize the polynomial method and simple random sampling. Our third result follows from a new dimensionality self reduction from the Orthogonal Vectors problem for $n$ vectors from $\{0,1\}^{d}$ to $n$ vectors from $\mathbb{Z}^{\ell}$ using Chinese Remainder Theorem, where $\ell = 2^{O(\log^{*} d)}$, dramatically improving the previous reduction in [Williams, SODA 2018].

We also establish a connection between conditional lower bounds for exact Max-IP with integer entries and NP $\cdot$ 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. Moreover, as a side product, we obtain an MA communication protocol for Set-Disjointness with complexity $O\left(\sqrt{n\log n \log\log n}\right)$, slightly improving the $O\left(\sqrt{n} \log n\right)$ bound [Aaronson and Wigderson, TOCT 2009], and approaching the $\Omega(\sqrt{n})$ lower bound [Klauck, CCC 2003].

Added discussion on related works.

Minor revisions.

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 $a \in A$ and $b \in B$ maximizing inner product $a \cdot b$. 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 $\ell_2$-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.

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 $n^{2 - \Omega(1)}$ time $\left( d/\log n \right)^{\Omega(1)}$-multiplicative-approximating algorithm, and we show this is conditionally optimal, as such a $\left(d/\log n\right)^{o(1)}$-approximating algorithm would refute SETH.

Characterization of Additive Approximation. 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 $n^{2 - \Omega(1)}$ time $\Omega(d)$-additive-approximating algorithm, and we show this is conditionally optimal, as such an $o(d)$-approximating algorithm would refute SETH.

$2^{O(\log^{*} n)}$-dimensional Hardness for Exact Max-IP Over The Integers. 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 $\mathbb{Z}^{d}$ for some $d = 2^{O(\log^{*} n)}$, every exact algorithm requires $n^{2 - o(1)}$ time. With the reduction from [Williams, SODA 2018], it follows that $\ell_2$-Furthest Pair and Bichromatic $\ell_2$-Closest Pair in $2^{O(\log^{*} n)}$ dimensions require $n^{2 - o(1)}$ time.

The lower bounds in our first and second results make use of a new MA protocol for Set-Disjointness introduced in [Rubinstein, 2017]. Our algorithms utilize the polynomial method and simple random sampling. Our third result follows from a new dimensionality self reduction from the Orthogonal Vectors problem for $n$ vectors from $\{0,1\}^{d}$ to $n$ vectors from $\mathbb{Z}^{\ell}$ using Chinese Remainder Theorem, where $\ell = 2^{O(\log^{*} d)}$, dramatically improving the previous reduction in [Williams, SODA 2018].

We also establish a connection between conditional lower bounds for exact Max-IP with integer entries and NP $\cdot$ 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. Moreover, as a side product, we obtain an MA communication protocol for Set-Disjointness with complexity $O\left(\sqrt{n\log n \log\log n}\right)$, slightly improving the $O\left(\sqrt{n} \log n\right)$ bound [Aaronson and Wigderson, TOCT 2009], and approaching the $\Omega(\sqrt{n})$ lower bound [Klauck, CCC 2003].