Improved Space Bounds for Subset Sum

TL;DR

The paper advances Subset Sum by delivering a tighter $O^*(2^{0.246n})$ space bound (with $O^*(2^{0.5n})$ time) through a streamlined algorithm that injects randomized prime filtering (à la Howgrave-Graham and Joux) into Schroeppel–Shamir’s scheme and leverages a representation technique to reduce to many Weighted Orthogonal Vector instances. It also presents a new Arthur–Merlin protocol framework for 4-SUM and, by reduction, for even $2k$-SUM, alongside conditional lower bounds for Circuit SAT that tie hardness of 4-SUM to circuit satisfiability. The work unifies modular arithmetic, probabilistic filtering, and mixer-based representations to achieve substantial space savings while preserving near-optimal time, and it clarifies the landscape of AM protocols and WOV-based reductions in this problem domain. The practical impact lies in enabling space-efficient exact subset-sum solutions for large instances, with implications for fine-grained complexity and certifiable subproblems via AM protocols.

Abstract

More than 40 years ago, Schroeppel and Shamir presented an algorithm that solves the Subset Sum problem for $n$ integers in time $O^*(2^{0.5n})$ and space $O^*(2^{0.25n})$. The time upper bound remains unbeaten, but the space upper bound has been improved to $O^*(2^{0.249999n})$ in a recent breakthrough paper by Nederlof and Węgrzycki (STOC 2021). Their algorithm is a clever combination of a number of previously known techniques with a new reduction and a new algorithm for the Orthogonal Vectors problem. In this paper, we improve the space bound by Nederlof and Węgrzycki to $O^*(2^{0.246n})$ and also simplify their algorithm and its analysis. We achieve this by using an idea, due to Howgrave-Graham and Joux, of using a random prime number to filter the family of subsets. We incorporate it into the algorithm by Schroeppel and Shamir and then use this amalgam inside the representation technique. This allows us to reduce an instance of Subset Sum to a larger number of instances of weighted orthogonal vector.

Figures (1)

• Figure 1: Partition of the instance $I$ (white) and its solution $S$ (gray) into parts.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Theorem 1.3
• Corollary 1.2
• Theorem 1.4
• Theorem 2.1: Chinese Remainder Theorem
• Lemma 2.1
• proof
• Definition 2.1