Counting Short Vector Pairs by Inner Product and Relations to the Permanent
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Given as input two n-element sets 𝒜,ℬ⊆{0,1}^d with d=clog n≤(log n)^2/(loglog n)^4 and a target t∈{0,1,…,d}, we show how to count the number of pairs (x,y)∈𝒜×ℬ with integer inner product ⟨ x,y ⟩=t deterministically, in n^2/2^Ω(√(log nloglog n/(clog^2 c))) 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 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.
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