Reading set notation for the flow number

Question

Could someone please help me to understand the following notation:

Flow Number:

In case of need, S refers to a set of flows, C(s) is called congestion and D(S) is dilation.

How would you put in words the notation that I have updated.

Thanks

Answer 1

$\min_S \max\{C(s), D(s)\}$ is a shortcut for $\min_{s ∈ S} \max\{C(s), D(s)\} = \min\{\max\{C(s), D(s)\}: s ∈ S\}$. In words: for each $s ∈ S$ i.e. for each flow $s$ you take whatever is bigger: its congestion $C(s)$ or dilatation $D(s)$; then from these chosen numbers you take the smallest.