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.    The two-party communication complexity of Boolean function f is known to be at least log rank (M  f  ), i.e., the logarithm of the rank of the communication matrix of f [19]. Lovász and Saks [17] asked whether the communication complexity of f can be bounded from above by (log rank (M  f  )) c  , for some constant c . The question was answered affirmatively for a special class of functions f in [17], and Nisan and Wigderson proved nice results related to this problem [20], but, for arbitrary f , it remained a difficult open problem.   We prove here an analogous polylogarithmic upper bound in the stronger multiparty communication model of Chandra et al. [6], which, instead of the rank of the communication matrix, depends on the L 1  norm of function f , for arbitrary Boolean function f .

Harmonic Analysis, Real Approximation, and the Communication Complexity of Boolean Functions