I would like to implement the multiple sink problem of Suurballe and Tarjan (https://doi.org/10.1002/net.3230140209) in the interpretation of Banerjee et al. (https://doi.org/10.1006/jpdc.1996.0035, relying only on Sections 2-3). However, I have multigraphs, for which the algorithm described in Sections 2-3 of Banerjee et al. does not work. Could you please help me how to reformulate the algorithm for multigraphs? Suurballe and Tarjan noted in page 334 that
"We can easily adapt our algorithm to work on arbitrary multidigraphs; that is, directed graphs with multiple edges, without the antisymmetry restriction. We need only redefine the tentative predecessor $p(u)$ of a vertex $u$ to be an edge entering $u$ rather than a vertex preceding $u$.",
so I think it is not a huge task to adapt this method for multigraphs, as well.
The easiest thing to do is to subdivide the edges causing you problems. That is, if there are multiple edges from $u$ to $v$ with costs $c_1, c_2, \dots, c_k$, then replace the $i^{\text{th}}$ edge with a pair of edges $uw_i, w_iv$, each with cost $\frac12 c_i$ (where $w_i$ is a new vertex created just for this edge and not used anywhere else).
Once this is done, you have a simple graph, and edge-disjoint paths in the new graph correspond to edge-disjoint paths in the old graph with the same cost.