Revisiting the Random Subset Sum problem

TL;DR

This paper provides a more elementary and direct proof for a known result about the randomised subset sum problem, showing that a small sample size suffices for high-probability approximations across a range.

Contribution

It offers an alternative, simpler proof of a key theorem in the random subset sum problem using elementary methods.

Findings

A new proof approach for the subset sum approximation theorem.

Confirmation that a logarithmic sample size suffices for high-probability approximations.

Enhanced understanding of the problem's properties through elementary techniques.

Abstract

The average properties of the well-known Subset Sum Problem can be studied by the means of its randomised version, where we are given a target value $z$, random variables $X_1, \ldots, X_n$, and an error parameter $\varepsilon > 0$, and we seek a subset of the $X_i$s whose sum approximates $z$ up to error $\varepsilon$. In this setup, it has been shown that, under mild assumptions on the distribution of the random variables, a sample of size $\mathcal{O}(\log(1/\varepsilon))$ suffices to obtain, with high probability, approximations for all values in $[-1/2, 1/2]$. Recently, this result has been rediscovered outside the algorithms community, enabling meaningful progress in other fields. In this work we present an alternative proof for this theorem, with a more direct approach and resourcing to more elementary tools.