In this paper, we provide simpler reductions from Exact Triangle to two important problems in fine-grained complexity: Exact Triangle with Few Zero-Weight $4$-Cycles and All-Edges Sparse Triangle. Exact Triangle instances with few zero-weight $4$-cycles was considered by Jin and Xu [STOC 2023], who used it as an intermediate problem to show $3$SUM hardness of All-Edges Sparse Triangle with few $4$-cycles (independently obtained by Abboud, Bringmann and Fischer [STOC 2023]), which is further used to show $3$SUM hardness of a variety of problems, including $4$-Cycle Enumeration, Offline Approximate Distance Oracle, Dynamic Approximate Shortest Paths and All-Nodes Shortest Cycles. We provide a simple reduction from Exact Triangle to Exact Triangle with few zero-weight $4$-cycles. Our new reduction not only simplifies Jin and Xu's previous reduction, but also strengthens the conditional lower bounds from being under the $3$SUM hypothesis to the even more believable Exact Triangle hypothesis. As a result, all conditional lower bounds shown by Jin and Xu [STOC 2023] and by Abboud, Bringmann and Fischer [STOC 2023] using All-Edges Sparse Triangle with few $4$-cycles as an intermediate problem now also hold under the Exact Triangle hypothesis. We also provide two alternative proofs of the conditional lower bound of the All-Edges Sparse Triangle problem under the Exact Triangle hypothesis, which was originally proved by Vassilevska Williams and Xu [FOCS 2020]. Both of our new reductions are simpler, and one of them is also deterministic -- all previous reductions from Exact Triangle or 3SUM to All-Edges Sparse Triangle (including P\u{a}tra\c{s}cu's seminal work [STOC 2010]) were randomized.

翻译：本文针对细粒度复杂性领域中的两个重要问题——"含少量零权4-环的精确三角形"与"全边稀疏三角形"——提出了源自精确三角形问题的简化归约方法。Jin 与 Xu [STOC 2023] 曾研究含少量零权4-环的精确三角形实例，将其作为中间问题证明含少量4-环的全边稀疏三角形问题的3SUM困难性（该结论由 Abboud、Bringmann 与 Fischer 独立获得 [STOC 2023]），并进一步推导出多种问题的3SUM困难性，包括4-环枚举、离线近似距离预言机、动态近似最短路径及全节点最短环。我们提出了一种从精确三角形到含少量零权4-环的精确三角形的简洁归约方法。该新归约不仅简化了 Jin 与 Xu 的既有方案，更将条件下界从基于3SUM假设强化为更可信的精确三角形假设。由此，Jin 与 Xu [STOC 2023] 及 Abboud、Bringmann 与 Fischer [STOC 2023] 通过含少量4-环的全边稀疏三角形作为中间问题所证明的所有条件下界，如今均在精确三角形假设下成立。此外，针对 Vassilevska Williams 与 Xu [FOCS 2020] 最初证明的全边稀疏三角形问题在精确三角形假设下的条件下界，我们提供了两种替代性证明。两项新归约均更为简洁，其中一项更具备确定性——此前所有从精确三角形或3SUM到全边稀疏三角形的归约（包括 Pătrașcu 的开创性工作 [STOC 2010]）均采用随机化方法。