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a constraint solver, they're used heavily in CAD as well

I imagine you could encode the problem as a CSP with each of the legal/contractual factors as a constraint. Despite NP-hardness of CSPs in general, CSP solvers are pretty efficient if the constraints are sparse enough and not chosen adversarially.

A basic start would be looking at network flow problems. While they don't extend to the complexity of real world airline scheduling, you can construct a simplified airline scheduling problem with it. Additionally, one of the methods to solve these problems, linear programming, does extend to integer linear programming which can be used to solve the more complicated cases (in non-deterministic polynomial time).