Harmonic analysis, real approximation, and the communication complexity of boolean functions

Abstract

The 2-party communication complexity of Boolean function f is known to be at least log rank (Mf), i.e. the logarithm of the rank of the communication matrix of f [17]. Lovász and Saks [15] asked whether the communication complexity of f can be bounded from above by (log rank (Mf))c, for some constant c. The question was answered affirmatively for a special class of functions f in [15], and Nisan and Wigderson proved nice results related to this problem [18], but for arbitrary f, it remained a difficult open problem. We prove here an analogous poly-logarithmic upper bound in the stronger multi-party communication model of Chandra, Furst and Lipton. [5], which, instead of the rank of the communication matrix, depends on the L1 norm of function f, for arbitrary Boolean function f.