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

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.