I have a graph with weighted edges. I want to remove edges to make the graph colorable with N colors (e.g. N=40) such that the total weight of removed edges is minimized. If I'm able to solve this problem, that will complete a project I've been working on for years now to make a working keyboard for a person I know that has cerebral palsy.
Regardless on the answer, I think this gonna be way too slow for your use case. Especially if you want the global optimum. Wikipedia says the current state-of-the-art is randomized algorithms, I don’t think these algorithms are looking for global optima.
I don’t recommend solving that, I think you’ll waste your time. How’s that related to the keyboard anyway?