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OV Graphs Are (Probably) Hard Instances

Authors Josh Alman, Virginia Vassilevska Williams

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Orthogonal Vectors
  • Fine-Grained Reductions
  • Cycle Finding

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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.

Cite As Get BibTex

Josh Alman and Virginia Vassilevska Williams. OV Graphs Are (Probably) Hard Instances. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 83:1-83:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ITCS.2020.83

Author Details

Josh Alman
  • Harvard University, Cambridge, MA, USA
Virginia Vassilevska Williams
  • MIT, Cambridge, MA, USA

Funding

  • Alman, Josh: Supported by a Michael O. Rabin Postdoctoral Fellowship.
  • Vassilevska Williams, Virginia: Partially supported by an NSF Career Award, a Sloan Fellowship, NSF Grants CCF-1417238, CCF-1528078 and CCF-1514339, and BSF Grant BSF:2012338.

Acknowledgements

The authors would like to thank the anonymous reviewers for their comments on an earlier version.

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