On the "log rank"-conjecture in communication complexity

We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between two n-element sets of vertices. Their goal is to decide whether or not the union of the two matcliings forms a Hamiltonian cycle. We prove: 1) The rank of the input matrix over the reals for this problem is 2^O(n). 2) The non-deterministic communication complexity of the problem is Ω(nloglog n). Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an Ω(nloglog n) lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument.