Towards a Theory of Multiparty Information Complexity

Communication complexity seeks to characterize the complexity of computation by measuring the amount of communication that must be used by players computing the output of some function f , each of whom have only partial information about the inputs to f . The past thirty years have seen the development of a number of lower-bound techniques in communication complexity, which fall roughly into two categories: rectangle-based techniques, which rely on combinatorial properties of the lookup table for f , and information-based techniques, which rely on bounds on the entropy in a communication protocol. Recently, [JK, BW12, KLL+12] have drawn a close-to-complete picture of the relationships among the various lower-bound techniques in the two-party communication model, concluding that information-based techniques are at least as good as almost all known rectangle-based techniques. Meanwhile, in the multiparty model, where, thus far, information bounds have not been used, it seems that new methods are required to prove nontrivial lower bounds when the number of players is large (logarithmic in the input size). Such bounds are important for one of the key applications of communication complexity— circuit lower bounds. Furthermore, it is conjectured that current techniques are insufficient to tightly characterize the complexity of some functions important both within communication complexity and for various applications, in particular the set disjointness function. In this thesis, we will investigate the possibility of extending information-based techniques to the case of more than two players. We prove the first nontrivial lower bounds on the information in a k-player protocol for k > 2, introducing a new sensitivity-based method to lower-bound information. We discuss progress towards extending our technique to randomized multiparty communication models.