Pigeonhole Equal Subset Sum in O ⃰ (2ⁿ/³)

Abstract

Subset Sum is a well known NP-hard problem. In Subset Sum, we are given a set of n integers S = {a1,··· ,an} and a target integer t, and are asked to find a subset A ⊆ [n] such that [formula]. We study a variant of the Subset Sum problem, Pigeonhole Equal Subset Sum. In Pigeonhole Equal Subset Sum, we are given a set of n integers S ={a₁,··· , aₙ} with the additional restriction that [formula], and want to find two different subsets A,B ⊆ [n] such that [formula]. The naive algorithm where we enumerate over all subset sums and look for a match takes O∗(2ⁿ) time. Horowitz and Sahni improve this to O ⃰ (2ⁿ/²) using a classical meet in the middle algorithm [1]. Recently, Jin and Wu improved this further to [formula] [2]. In this paper, we build on Jin and Wu’s techniques to improve the runtime even further to O ⃰ (2ⁿ/³).