Improving Schroeppel and Shamir’s algorithm for subset sum via orthogonal vectors

Abstract

We present an O∗(20.5n) time and O∗(20.249999n) space randomized algorithm for solving worst-case Subset Sum instances with n integers. This is the first improvement over the long-standing O∗(2n/2) time and O∗(2n/4) space algorithm due to Schroeppel and Shamir (FOCS 1979). We breach this gap in two steps: (1) We present a space efficient reduction to the Orthogonal Vectors Problem (OV), one of the most central problem in Fine-Grained Complexity. The reduction is established via an intricate combination of the method of Schroeppel and Shamir, and the representation technique introduced by Howgrave-Graham and Joux (EUROCRYPT 2010) for designing Subset Sum algorithms for the average case regime. (2) We provide an algorithm for OV that detects an orthogonal pair among N given vectors in {0,1}d with support size d/4 in time Õ(N· 2d/d d/4). Our algorithm for OV is based on and refines the representative families framework developed by Fomin, Lokshtanov, Panolan and Saurabh (J. ACM 2016). Our reduction uncovers a curious tight relation between Subset Sum and OV, because any improvement of our algorithm for OV would imply an improvement over the runtime of Schroeppel and Shamir, which is also a long standing open problem.