2026-08-05T22:58:39Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:11768 2024-03-06T10:48:40Z ddc:004 open_access

OV Graphs Are (Probably) Hard Instances Alman, Josh Vassilevska Williams, Virginia Orthogonal Vectors Fine-Grained Reductions Cycle Finding 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Josh Alman and Virginia Vassilevska Williams 2020 Is Part Of LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2020.83 urn:nbn:de:0030-drops-117686 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.83 eng https://creativecommons.org/licenses/by/3.0/legalcode