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

TL;DR

This paper presents a novel Subset Sum algorithm that improves upon the classical meet-in-the-middle approach, achieving faster worst-case running time in standard memory models.

Contribution

It introduces a new algorithm for worst-case Subset Sum with a running time better than the classical $O(2^{n/2})$, using combined advanced techniques.

Findings

Achieves worst-case time $O(2^{n/2} imes n^{- ext{constant}})$

First improvement over the classical meet-in-the-middle algorithm

Utilizes a combination of the representation method and bit-packing techniques.

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.