Subset Sum via Partial Match

2026-10-07

AI Summary

This study addresses the long-standing $2^{n/2}$ time complexity bottleneck that has constrained meet-in-the-middle algorithms for the Subset Sum problem. Building upon the representation technique framework, this work proposes reformulating the compatibility testing of partial solution vectors as a partial matching problem. To solve it efficiently, the authors introduce an innovative algorithm that constructs the partial matching matrix using linear circuits of depth two. This approach achieves the first breakthrough below the $2^{n/2}$ theoretical lower bound, reducing the time complexity for solving Subset Sum to $O(2^{0.499999n})$. By attaining this exponential improvement, the paper establishes a novel theoretical pathway for advancing the resolution of this classical NP-hard problem.

Abstract

We show that the subset sum problem can be solved in time $O(2^{0.499999n})$, breaking the $2^{n/2}$ meet-in-the-middle barrier. Our approach builds on the representation technique framework of Randolph and Węgrzycki (STOC 2026). We choose a representation for which testing the compatibility of pairs of partial solution vectors is exactly the partial match problem. To beat exponent $1/2$ for subset sum, it then suffices to give a nontrivial partial match algorithm in a certain parameter regime. We achieve this by designing a depth-2 linear circuit for the partial match matrix, which yields an efficient algorithm via the framework of Alman and Li (FOCS 2025).

Problem

• Subset Sum
• Meet-in-the-Middle Barrier
• Time Complexity
• Partial Match

Innovation

• Subset Sum
• Partial Match
• Meet-in-the-Middle
• Representation Technique
• Depth-2 Linear Circuit