 When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2020.83
URN: urn:nbn:de:0030-drops-117686
URL: https://drops.dagstuhl.de/opus/volltexte/2020/11768/
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### OV Graphs Are (Probably) Hard Instances

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### Abstract

A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v_1, …, v_n ∈ {0,1}^d such that nodes i and j are adjacent in G if and only if ⟨v_i,v_j⟩ = 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 ?_2. - 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.

### BibTeX - Entry

```@InProceedings{alman_et_al:LIPIcs:2020:11768,
author =	{Josh Alman and Virginia Vassilevska Williams},
title =	{{OV Graphs Are (Probably) Hard Instances}},
booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
pages =	{83:1--83:18},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-134-4},
ISSN =	{1868-8969},
year =	{2020},
volume =	{151},
editor =	{Thomas Vidick},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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