Improving Schroeppel and Shamir's Algorithm for Subset Sum via Orthogonal Vectors
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We present an ๐ช^โ(2^0.5n) time and ๐ช^โ(2^0.249999n) space randomized algorithm for solving worst-case Subset Sum instances with n integers. This is the first improvement over the long-standing ๐ช^โ(2^n/2) time and ๐ช^โ(2^n/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 ร(Nยท2^d/dd/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.
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