As one of the three main pillars of fine-grained complexity theory, the 3SUM problem explains the hardness of many diverse polynomial-time problems via fine-grained reductions. Many of these reductions are either directly based on or heavily inspired by P\u{a}tra\c{s}cu's framework involving additive hashing and are thus randomized. Some selected reductions were derandomized in previous work [Chan, He; SOSA'20], but the current techniques are limited and a major fraction of the reductions remains randomized. In this work we gather a toolkit aimed to derandomize reductions based on additive hashing. Using this toolkit, we manage to derandomize almost all known 3SUM-hardness reductions. As technical highlights we derandomize the hardness reductions to (offline) Set Disjointness, (offline) Set Intersection and Triangle Listing -- these questions were explicitly left open in previous work [Kopelowitz, Pettie, Porat; SODA'16]. The few exceptions to our work fall into a special category of recent reductions based on structure-versus-randomness dichotomies. We expect that our toolkit can be readily applied to derandomize future reductions as well. As a conceptual innovation, our work thereby promotes the theory of deterministic 3SUM-hardness. As our second contribution, we prove that there is a deterministic universe reduction for 3SUM. Specifically, using additive hashing it is a standard trick to assume that the numbers in 3SUM have size at most $n^3$. We prove that this assumption is similarly valid for deterministic algorithms.

翻译：作为细粒度复杂性理论三大支柱之一，3SUM问题通过细粒度归约揭示了众多多项式时间问题的内在难度。这些归约中许多直接基于或深受Pătrașcu的加性哈希框架启发，因而具有随机性。此前已有研究对部分归约进行了去随机化处理[Chan, He; SOSA'20]，但现有技术存在局限，大部分归约仍保持随机性。本文构建了一套旨在对基于加性哈希的归约进行去随机化的工具包。运用该工具包，我们成功对几乎所有已知的3SUM-困难归约实现了去随机化。作为技术亮点，我们特别对（离线）集合不相交性、（离线）集合交集与三角形列问题完成了难度归约的去随机化——这些问题在前人工作中被明确列为开放性问题[Kopelowitz, Pettie, Porat; SODA'16]。仅有的少数例外属于近期基于结构-随机性二分法的特殊归约类别。我们预期该工具包可便捷应用于未来归约的去随机化。作为概念创新，本工作推动了确定性3SUM-困难理论的发展。我们的第二个贡献在于证明了3SUM存在确定性的全集归约：通过加性哈希技术，通常可假设3SUM中数值大小不超过$n^3$，我们证明该假设对确定性算法同样成立。