A Quadratically Tight Partition Bound for Classical Communication and Query Complexity

The partition bound introduced in [4] is a way to prove lower bounds in classical communication and query complexity. While the partition bound provides a strong and general way to prove lower bounds, it remains open how tight the bounds obtained from this method are. In this work we give quadratically tight lower bounds via a strengthened version of the partition bound, which we call the public-coin partition bound. Formally, we show that, for all relations, the logarithms of the communication and query complexity versions of our public-coin partition bounds are within a quadratic factor of the public-coin randomized communication and randomized query complexity respectively.