Simpler Reductions from Exact Triangle
Abstract
In this paper, we provide simpler reductions from Exact Triangle to two important problems in fine-grained complexity: Exact Triangle with Few Zero-Weight -Cycles and All-Edges Sparse Triangle. Exact Triangle instances with few zero-weight -cycles was considered by Jin and Xu [STOC 2023], who used it as an intermediate problem to show SUM hardness of All-Edges Sparse Triangle with few -cycles (independently obtained by Abboud, Bringmann and Fischer [STOC 2023]), which is further used to show SUM hardness of a variety of problems, including -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 -cycles. Our new reduction not only simplifies Jin and Xu's previous reduction, but also strengthens the conditional lower bounds from being under the 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 -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.
Cite
@article{arxiv.2310.11575,
title = {Simpler Reductions from Exact Triangle},
author = {Timothy M. Chan and Yinzhan Xu},
journal= {arXiv preprint arXiv:2310.11575},
year = {2023}
}
Comments
To appear in SOSA 2024