2024/10/24 by Amir Abboud, Nick Fischer, Abboud, Amir +5
Computer Science · #Algorithms and Data Compression #Computational Complexity (cs.CC) #Constraint Satisfaction and Optimization #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2410.18661
openalex publication_date 2024/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(n1/4).