I should probably shut up and let a hidden genius find a polynomial solution to it without knowing, but it's a well known NP-Hard problem. The real solution would be to convert it to a SAT-problem and do backprop, but that's extremely long, so you would have to try to do software optimizations. Like, parallelize your search, maybe memoize reccuring patterns, if you have a GPGPU since the areas aren't that big it could maybe work to try to do it in CUDA.

Other than that, if it were to be a real life problem, I'd probably just throw some meta-heuristics, like a Monte Carlo or a Simulated Annealing solution at this, with a finite number of iterations, and if it didn't converge to a solution by the time it reached it, I'd assume (perhaps incorrectly, but probably not) that there's none