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Recognizing Sumsets is NP-Complete

Amir Abboud, Nick Fischer, Ron Safier, Nathan Wallheimer

Abstract

Sumsets are central objects in additive combinatorics. In 2007, Granville asked whether one can efficiently recognize whether a given set $S$ is a sumset, i.e. whether there is a set $A$ such that $A+A=S$. Granville suggested an algorithm that takes exponential time in the size of the given set, but can we do polynomial or even linear time? This basic computational question is indirectly asking a fundamental structural question: do the special characteristics of sumsets allow them to be efficiently recognizable? In this paper, we answer this question negatively by proving that the problem is NP-complete. Specifically, our results hold for integer sets and over any finite field. Assuming the Exponential Time Hypothesis, our lower bound becomes $2^{Ω(n^{1/4})}$.

Recognizing Sumsets is NP-Complete

Abstract

Sumsets are central objects in additive combinatorics. In 2007, Granville asked whether one can efficiently recognize whether a given set is a sumset, i.e. whether there is a set such that . Granville suggested an algorithm that takes exponential time in the size of the given set, but can we do polynomial or even linear time? This basic computational question is indirectly asking a fundamental structural question: do the special characteristics of sumsets allow them to be efficiently recognizable? In this paper, we answer this question negatively by proving that the problem is NP-complete. Specifically, our results hold for integer sets and over any finite field. Assuming the Exponential Time Hypothesis, our lower bound becomes .

Paper Structure

This paper contains 21 sections, 22 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Related Work
  4. Technical Overview
  5. Step 1: Encoding Assignments
  6. Step 2: Enforcing Satisfying Assignments
  7. Problem 1: Positioning
  8. Problem 2: Masking (And Stopping the Infinite Game)
  9. Additional Challenges for $\mathbb F_p^d$
  10. Conclusions and Open Questions
  11. Preliminaries
  12. Positioning
  13. Positioning for Z
  14. Positioning for Fpd
  15. Masking
  16. ...and 6 more sections

Key Result

theorem 1.2

The Sumset Recognition problem over $\mathbb Z$ is NP-complete.

Theorems & Definitions (23)

  • Definition 1.1: The Sumset Recognition Problem
  • theorem 1.2
  • theorem 1.3
  • lemma 1.3: Positioning for $\Int$
  • lemma 1.3: Masking for $\Int$
  • theorem 1.4: ETH-Hardness over $\mathbb Z$
  • lemma 2.1: Cauchy-Davenport Cauchy1813Davenport1935
  • lemma 3.0: Positioning for $\Int$
  • lemma 3.1: Skeleton Set for $\mathbb Z$
  • lemma 3.2
  • ...and 13 more