OV Graphs Are (Probably) Hard Instances

Authors Josh Alman, Virginia Vassilevska Williams

Thumbnail PDF


  • Filesize: 0.52 MB
  • 18 pages

Document Identifiers

Author Details

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


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

Cite AsGet 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)


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.

Subject Classification

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Amir Abboud and Virginia Vassilevska Williams. Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 434-443, 2014. Google Scholar
  2. Amir Abboud, Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms, pages 218-230. Society for Industrial and Applied Mathematics, 2015. Google Scholar
  3. Josh Alman, Timothy M Chan, and Ryan Williams. Polynomial representations of threshold functions and algorithmic applications. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 467-476. IEEE, 2016. Google Scholar
  4. Josh Alman and Ryan Williams. Probabilistic rank and matrix rigidity. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 641-652. ACM, 2017. Google Scholar
  5. Noga Alon. Covering graphs by the minimum number of equivalence relations. Combinatorica, 6(3):201-206, 1986. Google Scholar
  6. Noga Alon, Raphael Yuster, and Uri Zwick. Color-Coding. J. ACM, 42(4):844-856, 1995. Google Scholar
  7. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM Journal on Computing, 39(2):546-563, 2009. Google Scholar
  8. Lijie Chen and Ryan Williams. An equivalence class for orthogonal vectors. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 21-40. SIAM, 2019. Google Scholar
  9. Ruiwen Chen and Rahul Santhanam. Improved Algorithms for Sparse MAX-SAT and MAX-k-CSP. In Theory and Applications of Satisfiability Testing - SAT 2015 - 18th International Conference, Austin, TX, USA, September 24-27, 2015, Proceedings, pages 33-45, 2015. Google Scholar
  10. Marek Cygan, Marcin Pilipczuk, and Michał Pilipczuk. Known algorithms for edge clique cover are probably optimal. SIAM Journal on Computing, 45(1):67-83, 2016. Google Scholar
  11. Evgeny Dantsin and Alexander Wolpert. MAX-SAT for formulas with constant clause density can be solved faster than in o(s^2) time. In Theory and Applications of Satisfiability Testing - SAT 2006, 9th International Conference, Seattle, WA, USA, August 12-15, 2006, Proceedings, pages 266-276, 2006. Google Scholar
  12. Paul Erdös, Adolph W Goodman, and Louis Pósa. The representation of a graph by set intersections. Canadian Journal of Mathematics, 18:106-112, 1966. Google Scholar
  13. François Le Gall. Powers of tensors and fast matrix multiplication. In International Symposium on Symbolic and Algebraic Computation, ISSAC 2014, Kobe, Japan, July 23-25, 2014, pages 296-303, 2014. Google Scholar
  14. Jens Gramm, Jiong Guo, Falk Hüffner, and Rolf Niedermeier. Data reduction and exact algorithms for clique cover. Journal of Experimental Algorithmics (JEA), 13:2, 2009. Google Scholar
  15. András Gyárfás. A simple lower bound on edge coverings by cliques. Discrete Mathematics, 85(1):103-104, 1990. Google Scholar
  16. Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. Google Scholar
  17. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. Google Scholar
  18. Kasper Green Larsen and Ryan Williams. Faster online matrix-vector multiplication. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2182-2189. Society for Industrial and Applied Mathematics, 2017. Google Scholar
  19. Andrea Lincoln, Virginia Vassilevska Williams, and Ryan Williams. Tight hardness for shortest cycles and paths in sparse graphs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1236-1252. SIAM, 2018. Google Scholar
  20. László Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information theory, 25(1):1-7, 1979. Google Scholar
  21. László Lovász. Graphs and geometry. American Mathematical Society, 2019. Google Scholar
  22. László Lovász, Michael Saks, and Alexander Schrijver. Orthogonal representations and connectivity of graphs. Linear Algebra and its applications, 114:439-454, 1989. Google Scholar
  23. TS Michael and Thomas Quint. Sphericity, cubicity, and edge clique covers of graphs. Discrete Applied Mathematics, 154(8):1309-1313, 2006. Google Scholar
  24. Sylvia D Monson, Norman J Pullman, and Rolf Rees. A survey of clique and biclique coverings and factorizations of (0, 1)-matrices. Bull. Inst. Combin. Appl, 14:17-86, 1995. Google Scholar
  25. Salil P Vadhan et al. Pseudorandomness. Foundations and Trendsregistered in Theoretical Computer Science, 7(1-3):1-336, 2012. Google Scholar
  26. Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 887-898, 2012. Google Scholar
  27. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In Proceedings of the ICM, 2018. Google Scholar
  28. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2-3):357-365, 2005. Google Scholar
  29. Ryan Williams. Algorithms and resource requirements for fundamental problems. PhD thesis, Ph. D. Thesis, Carnegie Mellon University, CMU-CS-07-147, 2007. Google Scholar
  30. Raphael Yuster and Uri Zwick. Finding Even Cycles Even Faster. SIAM J. Discrete Math., 10(2):209-222, 1997. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail