For any given $(n,k,d)$ values, the MSR point $(\alpha_\text{MSR}, \beta_{\text{MSR}})$ of a helper scheme $A$ is defined by $$\begin{aligned}
\alpha_{\text{MSR}}\stackrel{\Delta}{=}\min \{\alpha: (\alpha,\beta)\text{ satisfies \eqref{eq:condition} and } \beta=\infty\}\label{eq:MSR-def}\\
\beta_{\text{MSR}}\stackrel{\Delta}{=}\min \{\beta: (\alpha,\beta)\text{ satisfies \eqref{eq:condition} and } \alpha=\alpha_{\text{MSR}}\}.\nonumber\end{aligned}$$

Specifically, the MBR and MSR are the two extreme ends[^3] of the bandwidth-storage tradeoff curve in .

By comparing and , we note that it is possible mathematically that when focusing on $\mathcal{G}_A$ ($\mathcal{G}_A$ is by definition a strict subset of $\mathcal{G}$) we may have $$\begin{aligned}
\min_{G\in \mathcal{G}_A}\min_{t\in \operatorname{DC}(G)}\operatorname{mincut}_G(s,t)> \min_{G\in \mathcal{G}}\min_{t\in \operatorname{DC}(G)}\operatorname{mincut}_G(s,t). \label{eq:outperform}\end{aligned}$$ If is true, then the given helper selection scheme $A$ strictly outperforms the BHS solution. Whether (or under what condition) is true is the main focus of this work.

Optimality and Weak Optimality of a Helper Selection Scheme {#sec:optimality}
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For future reference, we define the following optimality conditions.

\[def:scheme-optimal\] For any given $(n,k,d)$ value, a helper selection scheme $A$ is *absolutely optimal*, or simply *optimal*, if for any DHS scheme $B$ the following is true $$\begin{aligned}
\min_{G\in \mathcal{G}_A}\min_{t\in \operatorname{DC}(G)}\operatorname{mincut}_G(s,t)\geq \min_{G\in \mathcal{G}_B}\min_{t\in \operatorname{DC}(G)}\operatorname{mincut}_G(s,t)\nonumber\end{aligned}$$ for all $(\alpha,\beta)$ combinations. That is, scheme $A$ has the best $(\alpha,\beta)$ tradeoff curve among all DHS schemes and thus allows for the protection of the largest possible file 