Improving Schroeppel and Shamir's Algorithm for Subset Sum via Orthogonal Vectors

TL;DR

This paper introduces a faster randomized algorithm for the Subset Sum problem by reducing it to the Orthogonal Vectors problem and leveraging advanced techniques, achieving the first improvement over the classic Schroeppel-Shamir algorithm since 1979.

Contribution

It presents a novel reduction from Subset Sum to Orthogonal Vectors and develops an efficient OV algorithm, breaking the longstanding time and space complexity barriers.

Findings

Achieved an $ ilde{O}(2^{0.5n})$ time randomized algorithm for Subset Sum.

Developed an OV algorithm detecting orthogonal pairs in $ ilde{O}(N2^d/inom{d}{d/4})$ time.

Established a tight relation between Subset Sum and Orthogonal Vectors problems.

Abstract

We present an $\mathcal{O}^\star(2^{0.5n})$ time and $\mathcal{O}^\star(2^{0.249999n})$ space randomized algorithm for solving worst-case Subset Sum instances with $n$ integers. This is the first improvement over the long-standing $\mathcal{O}^\star(2^{n/2})$ time and $\mathcal{O}^\star(2^{n/4})$ 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 $\tilde{O}(N\cdot2^d/\binom{d}{d/4})$. 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.