Recognizing Sumsets is NP-Complete

TL;DR

This paper proves that recognizing whether a set is a sumset is NP-complete, indicating no efficient algorithm exists unless P=NP, and establishes exponential lower bounds under the ETH.

Contribution

It demonstrates the computational hardness of recognizing sumsets, resolving a long-standing open question and extending the NP-completeness result to various algebraic settings.

Findings

Recognition problem is NP-complete for integer sets and finite fields.

No polynomial-time algorithm is likely to exist for this problem.

Under ETH, the problem requires exponential time in the worst case.

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 $2^{\Omega(n^{1/4})}$.