A note on the partition bound for one-way classical communication complexity

TL;DR

This paper introduces a linear program for the one-way partition bound that precisely characterizes one-way randomized communication complexity with shared randomness, improving previous bounds by tightening the additive error term.

Contribution

It presents a new linear program for the one-way partition bound that offers a more accurate characterization of one-way randomized communication complexity with shared randomness.

Findings

Characterizes one-way randomized communication complexity using the partition bound.

Improves the previous rectangle bound characterization by reducing the additive error term.

Provides bounds relating the partition bound to communication complexity with shared randomness.

Abstract

We present a linear program for the one-way version of the partition bound (denoted $\mathsf{prt}^1_\varepsilon(f)$). We show that it characterizes one-way randomized communication complexity $\mathsf{R}_\varepsilon^1(f)$ with shared randomness of every partial function $f:\mathcal{X}\times\mathcal{Y}\to\mathcal{Z}$, i.e., for $\delta,\varepsilon\in(0,1/2)$, $\mathsf{R}_\varepsilon^1(f) \geq \log\mathsf{prt}_\varepsilon^1(f)$ and $\mathsf{R}_{\varepsilon+\delta}^1(f) \leq \log\mathsf{prt}_\varepsilon^1(f) + \log\log(1/\delta)$. This improves upon the characterization of $\mathsf{R}_\varepsilon^1(f)$ in terms of the rectangle bound (due to Jain and Klauck, 2010) by reducing the additive $O(\log(1/\delta))$-term to $\log\log(1/\delta)$.