XOR is not the correct name, but I am looking for some kind of exclusive behavior.
I am currently solving a set of different (assignment) problems by modeling flow networks and running a min-cost-max-flow algorithm. Flow networks are quite handy because a lot of problems can be reduced to them in an easy and understandable way. In my case these are matchings with some additional constraints. As these constraints are getting more complex I've been wondering if there are some existing constructions to model specific behaviors.
In this case I want to restrict the outgoing flow of a node to a single edge.
Given a graph $G=(V, E)$, integral capacities $c(u,v)$ and costs $k(u,v)$. An arbitrary node is called $A$. It's direct neighbors are called $B_1, ..B_n$. Can we replace the edges $AB_1,...AB_n$ (red) with some construction so that only one edge can receive flow? Which means that if $AB_1$ gets some flow (e.g. $5/10$) no other (red) edge can receive flow.
We could add intermediate nodes/edges and play with costs and capacities. The total capacity of our new construction has to stay the same and the cost of the different alternatives have to stay somehow proportional.
So my questions are:
- Are there constructions like this in general? (Any keywords, links, papers)
- Can you suggest a solution to my specific problem?