A comparison of two lower bound methods for communication complexity

Abstract

The methods “Rank” and “Fooling Set” for proving lower bounds on the deterministic communication complexity of Boolean functions are compared. The main results are as follows. (i) The Rank method provides the lower bound n on communication complexity for almost all Boolean functions of 2n variables, whereas the Fooling Set method provides only the lower bound d(n) ≤ log2n + log2 10. A specific sequence of Boolean functions {f2n}∞n=1, f2n of 2n variables, is constructed, such that the Rank method provides exponentially higher lower bounds for f2n than the Fooling Set method. (ii) A specific sequence of Boolean functions {f2n}∞n=1 is constructed such that the Fooling Set method provides a lower bound of n for h2n, whereas the Rank method provides only (log2 3)/2 · n ≈ 0.79 · n as a lower bound. (iii) It is proved that lower bounds obtained by the Fooling Set method are better by at most a factor of two compared with lower bounds obtained by the Rank method. These three results together solve the last problem about the comparison of lower bound methods on communication complexity left open in (Aho,A.V., Ullman, J.D., Yannakakis, M., On notions of information transfer in VLSI circuits, in: Proc. 15th ACM STOC 1983, pp. 133–139). Finally, it is shown that an extension of the Fooling Set method provides lower bounds which are tight (up to a polynomial) for all Boolean functions.