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

翻译：和集是加性组合数学中的核心研究对象。2007年，Granville提出一个基本计算问题：能否高效判定给定集合$S$是否为和集，即是否存在集合$A$使得$A+A=S$。Granville提出了一种算法，其时间复杂度相对于给定集合的大小呈指数级，但我们能否实现多项式时间甚至线性时间的算法？这一计算问题实质上指向一个更深层的结构性问题：和集所具有的特殊性质是否使其能够被高效识别？本文通过证明该问题是NP完全的，对此给出了否定回答。具体而言，我们的结论对整数集及任意有限域上的集合均成立。若假设指数时间假说成立，我们得到的下界为$2^{\Omega(n^{1/4})}$。