Let's assign b1 = −∞.
After that simple reduction, what you describe is exactly the maximum blocking-cut problem, for example defined in "The Pseudoflow Algorithm: A New Algorithm for the Maximum-Flow Problem", Hochbaum 2008:
Given a directed graph G = (V, A), node weights positive or negative wi for all i ∈ V, and non-negative arc weights cij for all (i,j) ∈ A. [...] Find a subset of nodes S ⊆ V such that
is maximal.
They continue to draw a connection between the maximum blocking-cut problem and the min-cut problem. For that, the define V' = V ∪ { s,t } and A' = A ∪ { (s,i) | wi > 0 } ∪ { (j,t) | wj < 0 }. Then they define the arc capacities csi = wi if wi > 0 and cjt = −wj if wj < 0. We then have
For S ⊆ V, {s} ∪ S is the source set of a minimum cut in Gst = (V', A') if and only if (S, V \ S) is a maximum blocking cut in the graph G.
The latter problem can be solved easily using a maximum-flow algorithm, by utilizing the max-flow min-cut theorem. In fact you can use job 1 as the source since it is guaranteed to be part of the source set (the jobs running on the old machine).