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{\'e}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.

翻译：我们的工作探讨了3SUM实例在缺乏特定加法结构时的难度及其应用。作为主要技术成果，我们证明：在避免满足$a+b=c+d$（其中$\{a, b\} \ne \{c, d\}$）的大小为$n$的整数集合上求解3SUM问题，在3SUM假设下仍需要$n^{2-o(1)}$时间。这类集合被称为Sidon集，在加法组合领域已有深入研究。— 结合先前归约，这意味着：对于最大度为$\sqrt{n}$且对任意$k \ge 3$最多包含$n^{k/2}$个$k$-环的$n$顶点图上的全边稀疏三角问题，在3SUM假设下仍需要$n^{2-o(1)}$时间。这可用于加强Abboud、Bringmann、Khoury和Zamir [STOC'22] 关于4-环枚举、离线近似距离查询和近似动态最短路径问题的先前条件性下界。特别地，我们证明：对于$n$顶点$m$边且允许$n^{o(1)}$延迟的4-环枚举问题，不存在预处理时间为$O(n^{2-\varepsilon})$或$O(m^{4/3-\varepsilon})$（其中$\varepsilon >0$）的算法。我们还通过对已知4-环检测算法进行简单修改，给出了匹配的上界。— 主要结论的轻微推广还扩展了Dudek、Gawrychowski和Starikovskaya [STOC'20] 关于非平凡3变量线性退化测试（3-LDT）的3SUM困难性结果：我们证明了所有非平凡4-LDT的3SUM困难性。主要技术结论的证明综合运用了多种工具：Balog-Szemerédi-Gowers定理、稀疏卷积算法，以及一种新的、对无小系数线性关系的整数具有几乎3-通用保证的近似线性哈希函数。