Fredman’s Trick Meets Dominance Product: Fine-Grained Complexity of Unweighted APSP, 3SUM Counting, and More

Chan, Timothy M.; Williams, Virginia Vassilevska; Xu, Yinzhan

doi:10.1145/3564246.3585237

Cited by 3 publications

(4 citation statements)

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“…Our new reduction is inspired by the recent work of Chan, Vassilevska Williams and Xu [CVX23]. In particular, they showed versions of the Balog-Szemerédi-Gowers theorem that have simpler proofs inspired by the famous "Fredman's trick" 2 [Fre76].…”

Section: Sum With Moderate Energymentioning

confidence: 99%

“…Now the problem becomes the following: for every (i, k) ∈ E 0 , test whether there exists j such that A ij = B jk . This can be solved by computing the so-called Equality Product of the two matrices A and B, which is known to be in O(n (3+ω)/2 ) time [Mat91] (see [CVX23] for more applications of Equality Product combined with Fredman's trick). After this step, we can remove E 0 from E, and by symmetry, we can also remove all edges that are reverses to the edges in E 0 .…”

Section: Exact Triangle With Few Zero-weight 4-cyclesmentioning

confidence: 99%

“…Remark A.4. Our first approach in Section 3 can also be adapted to give a direct reduction from 3SUM to All-Edges Sparse Triangle, by combining the idea of halving the weights with the reduction from Real 3SUM to #All-Edges Sparse Triangle in [CVX23]. The resulting reduction is deterministic and completely avoids hashing, unlike all previous reductions.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…However, we have the following result if we assume the Strong Exact Triangle hypothesis, namely, that Exact Triangle for integer weights in [±n] requires n 3−o(1) time. (Recent work has considered similarly defined Strong APSP hypothesis, Strong 3SUM hypothesis, and so on[CVX23,JX23].) The proof here only needs Lemma 4.1, so the reduction is even simpler (and deterministic).…”

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Hardness for triangle problems under even more believable hypotheses: reductions from real APSP, real 3SUM, and OV

Chan

Williams

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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The 3SUM hypothesis, the All-Pairs Shortest Paths (APSP) hypothesis and the Strong Exponential Time Hypothesis are the three main hypotheses in the area of fine-grained complexity. So far, within the area, the first two hypotheses have mainly been about integer inputs in the Word RAM model of computation. The "Real APSP" and "Real 3SUM" hypotheses, which assert that the APSP and 3SUM hypotheses hold for real-valued inputs in a reasonable version of the Real RAM model, are even more believable than their integer counterparts.Under the very believable hypothesis that at least one of the Integer 3SUM hypothesis, Integer APSP hypothesis or SETH is true, Abboud, Vassilevska W. and Yu [STOC 2015] showed that a problem called Triangle Collection requires 𝑛 3−𝑜 (1) time on an 𝑛-node graph.The main result of this paper is a nontrivial lower bound for a slight generalization of Triangle Collection, called All-Color-Pairs Triangle Collection, under the even more believable hypothesis that at least one of the Real 3SUM, the Real APSP, and the Orthogonal Vector (OV) hypotheses is true. Combined with slight modifications of prior reductions from Triangle Collection, we obtain polynomial conditional lower bounds for problems such as the (static) ST-Max Flow problem and dynamic versions of Max Flow, Single-Source Reachability Count, and Counting Strongly Connected Components, now under the new weaker hypothesis.Our main result is built on the following two lines of reductions. In the first line of reductions, we show Real APSP and Real 3SUM hardness for the All-Edges Sparse Triangle problem. Prior reductions only worked from the integer variants of these problems. In the second line of reductions, we show Real APSP and OV hardness for a variant of the Boolean Matrix Multiplication problem.

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Section: Sum With Moderate Energymentioning

confidence: 99%

Section: Exact Triangle With Few Zero-weight 4-cyclesmentioning

confidence: 99%

Section: Acknowledgementsmentioning

confidence: 99%

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