Non-deterministic communication complexity with few witnesses

Abstract

We study non-deterministic communication protocols in which no input has too many witnesses. Define n k ( f ) to be the minimum complexity of a non-deterministic protocol for the function f in which each input has at most k witnesses. We present two different lower bounds for n k ( f ). Our first result shows that n k ( f ) is below by Ω (√ c ( f )/ k ), where c ( f ) is the deterministic complexity. Our second results bounds n k ( f ) by log(rk( M f ))/ k − 1, where rk( M f ) is the rank of the representing matrix of f . As a consequence, it follows that the communication complexity analogue of the Turing-complexity class FewP is equal to the analogue of the class P .