Subset Sum in Time 2^n/2 / poly(n)

01/17/2023
by   Xi Chen, et al.
0

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· n^-γ) for a constant γ > 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.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset