Matching Triangles and Basing Hardness on an Extremely Popular Conjecture
Name
15m1050987.pdf
Description
Published version
Size
443.27 KB
Format
Adobe PDF
Checksum (MD5)
47943e13ab4b549ea07e6470a845b93e
Author(s)
Abboud, Amir
Williams, Virginia Vassilevska
Yu, Huacheng
Date Issued
January 2018
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Abboud, Amir, Williams, Virginia Vassilevska and Yu, Huacheng. 2018. "Matching Triangles and Basing Hardness on an Extremely Popular Conjecture." 47 (3).
Version
Final published version
Abstract
© 2018 Society for Industrial and Applied Mathematics. Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the all pairs shortest paths (APSP) conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining “less conditional” and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including tight n3−o(1) lower bounds for purely combinatorial problems about the triangles in unweighted graphs; new n1−o(1) lower bounds for the amortized update and query times of dynamic algorithms for Single-Source Reachability, Strongly Connected Components, and Max-Flow; new n1.5−o(1) lower bound for computing a set of n st-maximum-flow values in a directed graph with n nodes and Õ(n) edges; and a hierarchy of natural graph problems on n nodes with complexity nc for c ∈ (2, 3). Only slightly nontrivial consequences of this conjecture were known prior to our work. Along the way we also obtain new conditional lower bounds for the Single-Source Max-Flow problem.
Subjects
General Mathematics
General Computer Science
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1137/15m1050987
