OV graphs are (probably) hard instances
Name
LIPIcs-ITCS-2020-83.pdf
Description
Published version
Size
537.49 KB
Format
Adobe PDF
Checksum (MD5)
bc8152bbba02b0e0998ce88d772d582e
Author(s)
Alman, Josh
Williams, Virginia Vassilevska
Date Issued
2020
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Citation
Alman, Josh and Williams, Virginia Vassilevska. 2020. "OV graphs are (probably) hard instances." Leibniz International Proceedings in Informatics, LIPIcs, 151.
Version
Final published version
Abstract
© Josh Alman and Virginia Vassilevska Williams. A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v1, . . ., vn ∈ {0, 1}d such that nodes i and j are adjacent in G if and only if hvi, vji = 0 over Z. In this paper, we study a number of basic graph algorithm problems, except where one is given as input the vectors defining an OV graph instead of a general graph. We show that for each of the following problems, an algorithm solving it faster on such OV graphs G of dimension only d = O(log n) than in the general case would refute a plausible conjecture about the time required to solve sparse MAX-k-SAT instances: Determining whether G contains a triangle. More generally, determining whether G contains a directed k-cycle for any k ≥ 3. Computing the square of the adjacency matrix of G over Z or F2. Maintaining the shortest distance between two fixed nodes of G, or whether G has a perfect matching, when G is a dynamically updating OV graph. We also prove some complementary results about OV graphs. We show that any problem which is NP-hard on constant-degree graphs is also NP-hard on OV graphs of dimension O(log n), and we give two problems which can be solved faster on OV graphs than in general: Maximum Clique, and Online Matrix-Vector Multiplication.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Creative Commons Attribution 3.0 unported license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.4230/LIPIcs.ITCS.2020.83
