Skip to main content

On rank vs. communication complexity


This paper concerns the open problem of Lovász and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We first give an example exhibiting the largest gap known. We then prove two related theorems.

This is a preview of subscription content, access via your institution.


  1. [1]

    N. Alon, P. Seymour: A counterexample to the rank-covering conjectureJ. Graph Theory,13, (1989), 523–525.

    Google Scholar 

  2. [2]

    S. Fajtlowicz: On conjectures of Graffiti II,Congresus Numeratum 60 (1987), 189–198.

    Google Scholar 

  3. [3]

    B. Kalyanasundaram andG. Schnitger: The probabilistic communication complexity of set intersection,2nd Structure in Complexity Theory Conference, (1987), 41–49.

  4. [4]

    E. Kushilevitz: private communication, 1994.

  5. [5]

    L. Lovász: Communication Complexity: A survey, in:Paths, Flows, and VLSI Layout, B. H. Korte, ed., Springer Verlag, Berlin 1990.

    Google Scholar 

  6. [6]

    L. Lovász andM. Saks: Lattices, Möbius functions, and communication complexity,Proc. of the 29th FOCS, (1988), 81–90.

  7. [7]

    L. Lovász andM. Saks: Private communication.

  8. [8]

    K. Mehlhorn, E. M. Schmidt: Las Vegas is better than determinism in VLSI and distributive computing,Proceedings of 14th STOC, (1982), 330–337.

  9. [9]

    N. Nisan andM. Szegedy: On the degree of boolean functions as real polynomials,Proceedings of 24th STOC, (1992), 462–467.

  10. [10]

    N. Nisan andA. Wigderson: On rank vs. communication complexity,Proceedings of 35th FOCS, (1994), 831–836.

  11. [11]

    C. van Nuffelen: A bound for the chromatic number of graph,American Mathematical Monthly 83, (1976), 265–266.

    Google Scholar 

  12. [12]

    A. Razborov: On the distributional complexity of disjointness,Theoretical Computer Science 106 (1992), 385–390.

    Google Scholar 

  13. [13]

    A. Razborov, The gap between the chromatic number of a graph and the rank of its adjacency matrix is superlinear,Discrete Math.,108, (1992), 393–396.

    Google Scholar 

  14. [14]

    R. Raz andB. Spiker: On the Log-Rank conjecture in communication complexity,Proc. of the 34th FOCS, (1993), 168–176;Combinatorica 15(4), (1995), 567–588.

  15. [15]

    A. C.-C. Yao: Some complexity questions related to distributive computing.Proceedings of 11th STOC, (1979), 209–213.

  16. [16]

    A. C.-C. Yao: Lower Bounds by Probabilistic Arguments,Proc. 24th FOCS, (1983), 420–428.

Download references

Author information



Additional information

A preliminary version of this paper appeared in [10].

This work was supported by USA-Israel BSF grant 92-00043 and by a Wolfeson research award administered by the Israeli Academy of Sciences.

This work was supported by USA-Israel BSF grant 92-00106 and by a Wolfeson research award administered by the Israeli Academy of Sciences.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Nisan, N., Wigderson, A. On rank vs. communication complexity. Combinatorica 15, 557–565 (1995).

Download citation

Mathematics Subject Classification (1991)

  • 68Q05
  • 68R05
  • 05C50