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On the “log rank”-conjecture in communication complexity

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2 Author(s)
Raz, R. ; Dept. of Comput. Sci., Princeton Univ., NJ, USA ; Spieker, B.

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 matchings forms a Hamiltonian cycle. We prove: (1) The rank of the input matrix over the reals for this problem is 2O(n). (2) The non-deterministic communication complexity of the problem is Ω(n log log 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(n log log 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

Published in:

Foundations of Computer Science, 1993. Proceedings., 34th Annual Symposium on

Date of Conference:

3-5 Nov 1993