Counting Short Vector Pairs by Inner Product and Relations to the Permanent

TL;DR

This paper presents a deterministic subquadratic algorithm for counting vector pairs with a given inner product, nearly matching randomized bounds, and explores its connection to computing the permanent of a zero-one matrix.

Contribution

It introduces a novel deterministic algorithm for counting vector pairs by inner product using prime residue techniques and relates this problem's complexity to computing the matrix permanent.

Findings

Deterministic algorithm runs in near subquadratic time for certain dimensions.

The method extends to randomized algorithms with high probability guarantees.

Establishes a connection between inner product counting and matrix permanent computation.

Abstract

Given as input two $n$-element sets $\mathcal A,\mathcal B\subseteq\{0,1\}^d$ with $d=c\log n\leq(\log n)^2/(\log\log n)^4$ and a target $t\in \{0,1,\ldots,d\}$, we show how to count the number of pairs $(x,y)\in \mathcal A\times \mathcal B$ with integer inner product $\langle x,y \rangle=t$ deterministically, in $n^2/2^{\Omega\bigl(\!\sqrt{\log n\log \log n/(c\log^2 c)}\bigr)}$ time. This demonstrates that one can solve this problem in deterministic subquadratic time almost up to $\log^2 n$ dimensions, nearly matching the dimension bound of a subquadratic randomized detection algorithm of Alman and Williams [FOCS 2015]. We also show how to modify their randomized algorithm to count the pairs w.h.p., to obtain a fast randomized algorithm. Our deterministic algorithm builds on a novel technique of reconstructing a function from sum-aggregates by prime residues, which can be seen as an {\em additive} analog of the Chinese Remainder Theorem. As our second contribution, we relate the fine-grained complexity of the task of counting of vector pairs by inner product to the task of computing a zero-one matrix permanent over the integers.