I have a question regarding a problem I'm working on.
The problem is given an MxN grid with k sources and sinks, find non intersecting paths (vertex disjoint) such that all sources are paired with a sink. Each source must be paired to a single sink, and each sink must be paired to a single source, however all of the sources are compatible with all of the sinks. This could be imagined as k identical locks and k identical keys, all that matters is that its possible to move the keys into locks without having paths that intersect. By contrast in the original problem, sink Si must be paired with source Di and this is an input parameter of the problem.
The underlying problem is NP-Hard but according to a vague comment I found, with the given restriction it can be solved in polynomial time. Unfortunately I can't find any relevant literature that could help.
If anyone could point me towards proofs or papers that discuss this problem I would appreciate it.