Removing Additive Structure in 3SUM-Based Reductions

Our work explores the hardness of \(3\)SUM instances without certain additive structures, and its applications. As our main technical result, we show that solving \(3\)SUM on a size-\(n\) integer set that avoids solutions to \(a+b=c+d\) for \(\{a, b\} \ne \{c, d\}\) still requires \(n^{2-o(1)}\) time, under the \(3\)SUM 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 \(n\)-vertex graphs with maximum degree \(\sqrt{n}\) and at most \(n^{k/2}\) \(k\)-cycles for every \(k \ge 3\) requires \(n^{2-o(1)}\) time, under the \(3\)SUM hypothesis. This can be used to strengthen the previous conditional lower bounds by Abboud, Bringmann, Khoury, and Zamir [STOC'22] 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 \(n\)-vertex \(m\)-edge graphs with \(n^{o(1)}\) delays has \(O(n^{2-\varepsilon})\) or \(O(m^{4/3-\varepsilon})\) pre-processing time for \(\varepsilon >0\). 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 [STOC'20] on the \(3\)SUM hardness of nontrivial 3-Variate Linear Degeneracy Testing (3-LDTs): we show \(3\)SUM hardness for all nontrivial 4-LDTs. The proof of our main technical result combines a wide range of tools: Balog-Szemer{é}di-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.