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Revisiting the Random Subset Sum problem

Arthur da Cunha, Francesco d'Amore, Frédéric Giroire, Hicham Lesfari, Emanuele Natale, Laurent Viennot

TL;DR

An alternative proof for this theorem is presented, with a more direct approach and resourcing to more elementary tools, that suffices to obtain approximations for all values in $[-1/2, 1/2]$.

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.

Revisiting the Random Subset Sum problem

TL;DR

An alternative proof for this theorem is presented, with a more direct approach and resourcing to more elementary tools, that suffices to obtain approximations for all values in .

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 , random variables , and an error parameter , and we seek a subset of the s whose sum approximates up to error . In this setup, it has been shown that, under mild assumptions on the distribution of the random variables, a sample of size suffices to obtain, with high probability, approximations for all values in . 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.
Paper Structure (7 sections, 7 theorems, 32 equations)

This paper contains 7 sections, 7 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Our argument
  3. Preliminaries
  4. Expected behaviour
  5. Growth of the volume up to 1/2
  6. Growth of the volume from 1/2
  7. Putting everything together

Key Result

Theorem 1

Let $X_1, \dots, X_n$ be independent uniform random variables over $[-1, 1]$, and let $\varepsilon \in (0, 1/3)$. There exists a universal constant $C > 0$ such that, if $n \ge C\log(1/\varepsilon)$, then, with probability at least $1 - \varepsilon$, for all $z \in [-1, 1]$ there exists $S_z \subset

Theorems & Definitions (12)

  • Theorem 1: Lueker, 1998
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary 4
  • Lemma 5
  • proof
  • ...and 2 more