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arXiv · 2301.07134

Subset Sum in Time $2^{n/2} / poly(n)$

Xi Chen·Yaonan Jin·Tim Randolph·Rocco A. Servedio
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Abstract

A major goal in the area of exact exponential algorithms is to give an algorithm for the (worst-case) $n$-input Subset Sum problem that runs in time $2^{(1/2 - c)n}$ for some constant $c>0$. In this paper we give a Subset Sum algorithm with worst-case running time $O(2^{n/2} \cdot n^{-\gamma})$ for a constant $\gamma > 0.5023$ in standard word RAM or circuit RAM models. To the best of our knowledge, this is the first improvement on the classical ``meet-in-the-middle'' algorithm for worst-case Subset Sum, due to Horowitz and Sahni, which can be implemented in time $O(2^{n/2})$ in these memory models. Our algorithm combines a number of different techniques, including the ``representation method'' introduced by Howgrave-Graham and Joux and subsequent adaptations of the method in Austrin, Kaski, Koivisto, and Nederlof, and Nederlof and Wegrzycki, and ``bit-packing'' techniques used in the work of Baran, Demaine, and Patrascu on subquadratic algorithms for 3SUM.

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Xi Chen, Yaonan Jin, Tim Randolph, Rocco A. Servedio. 2023-01-17. Subset Sum in Time $2^{n/2} / poly(n)$. https://arxiv.org/abs/2301.07134

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