Removing Additive Structure in 3SUM-Based Reductions

SIAM Journal on Computing, Ahead of Print.
Abstract. Our work explores the hardness of 3SUM instances without certain additive structures, and its applications. As our main technical result, we show that solving 3SUM on a size-[math] integer set that avoids solutions to [math] for [math] still requires [math] time, under the 3SUM hypothesis. Such sets are called Sidon sets and are well-studied in the field of additive combinatorics. Combined with previous reductions, this implies that the all-edges sparse triangle problem on [math]-vertex graphs with maximum degree [math] and at most [math] [math]-cycles for every [math] requires [math] time, under the 3SUM hypothesis. This can be used to strengthen the previous conditional lower bounds by Abboud, Bringmann, Khoury, and Zamir [54th Annual ACM SIGACT Symposium on Theory of Computing, 2022] of 4-cycle enumeration, offline approximate distance oracle and approximate dynamic shortest path. In particular, we show that no algorithm for the 4-cycle enumeration problem on [math]-vertex [math]-edge graphs with [math] delays has [math] or [math] preprocessing time for [math]. We also present a matching upper bound via simple modifications of the known algorithms for 4-cycle detection. A slight generalization of the main result also extends the result of Dudek, Gawrychowski, and Starikovskaya [52nd Annual ACM SIGACT Symposium on Theory of Computing, 2020] on the 3SUM-hardness of nontrivial 3-variate linear degeneracy testing (3-LDTs): we show 3SUM-hardness for all nontrivial 4-LDTs. The proof of our main technical result combines a wide range of tools: Balog–Szemerédi–Gowers theorem, sparse convolution algorithm, and a new almost-linear hash function with almost 3-universal guarantee for integers that do not have small-coefficient linear relations.