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.

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  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.

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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.

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Nisan, N., Wigderson, A. On rank vs. communication complexity. Combinatorica 15, 557–565 (1995). https://doi.org/10.1007/BF01192527

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Mathematics Subject Classification (1991)

  • 68Q05
  • 68R05
  • 05C50