Communication Complexity and Quasi Randomness

The multiparty communication complexity concerns the least number of bits that must be exchanged among a number of players to collaboratively compute a Boolean function $f ( x_1 , \ldots ,x_k )$, while each player knows at most t inputs for some fixed $t < k$. The relation of the multiparty communication complexity to various hypergraph properties is investigated. Many of these properties are satisfied by random hypergraphs and can be classified by the framework of quasi randomness. Namely, many disparate properties of hypergraphs are shown to be mutually equivalent, and, furthermore, various equivalence classes form a natural hierarchy. In this paper, it is proved that the multiparty communication complexity problems are equivalent to certain hypergraph properties and thereby establish the connections among a large number of combinatorial and computational aspects of hypergraphs or Boolean functions.