Recognizing Sumsets is NP-Complete

Amir Abboud; Nick Fischer; Ron Safier; Nathan Wallheimer

doi:https://doi.org/10.1137/1.9781611978322.153

Recognizing Sumsets is NP-Complete

Amir Abboud, Nick Fischer, Ron Safier, Nathan Wallheimer

Department of Computer Science and Applied Mathematics

Weizmann Institute of Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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.

Original language	English
Title of host publication	Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025
Pages	4484-4506
Number of pages	23
ISBN (Electronic)	9798331312008
DOIs	https://doi.org/10.1137/1.9781611978322.153
State	Published - Jan 2025
Event	36th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025 - New Orleans, United States Duration: 12 Jan 2025 → 15 Jan 2025

Publication series

Name	Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume	7
ISSN (Print)	1071-9040

Conference

Conference	36th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025
Country/Territory	United States
City	New Orleans
Period	12/01/25 → 15/01/25

All Science Journal Classification (ASJC) codes

Software
General Mathematics

Access to Document

https://doi.org/10.1137/1.9781611978322.153

http://arxiv.org/abs/2410.18661License: CC BY

Cite this

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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.",

author = "Amir Abboud and Nick Fischer and Ron Safier and Nathan Wallheimer",

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Abboud, A, Fischer, N, Safier, R & Wallheimer, N 2025, Recognizing Sumsets is NP-Complete. in Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, vol. 7, pp. 4484-4506, 36th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, United States, 12/01/25. https://doi.org/10.1137/1.9781611978322.153

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AU - Safier, Ron

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PY - 2025/1

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N2 - 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.

AB - 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.

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M3 - منشور من مؤتمر

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

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BT - Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025

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Y2 - 12 January 2025 through 15 January 2025

ER -

Recognizing Sumsets is NP-Complete

Abstract

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