The Partition Bound for Classical Communication Complexity and Query Complexity

TL;DR

This paper introduces the partition bound, a new linear-program-based method for establishing lower bounds in randomized communication and query complexity, surpassing many existing bounds.

Contribution

The paper presents the partition bound as a novel, stronger lower bound technique for communication and query complexity, unifying and improving upon previous bounds.

Findings

Partition bound outperforms rectangle/corruption and _ bounds in communication complexity.

Partition bound exceeds approximate polynomial degree and adversary bounds in query complexity.

Example showing quadratic gap between partition bound and polynomial degree.

Abstract

We describe new lower bounds for randomized communication complexity and query complexity which we call the partition bounds. They are expressed as the optimum value of linear programs. For communication complexity we show that the partition bound is stronger than both the rectangle/corruption bound and the \gamma_2/generalized discrepancy bounds. In the model of query complexity we show that the partition bound is stronger than the approximate polynomial degree and classical adversary bounds. We also exhibit an example where the partition bound is quadratically larger than polynomial degree and classical adversary bounds.