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

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 $\log\log(1/\delta)$.