This is a very popular article that get submitted every now and then (nearly every year) [1].

I think this kind of problem would be a very nice for logic, optimization and constraint programming that probably can be solved with modern tools like Google OR-Tool or Monash University MiniZinc [1],[2],[3].

[1] Past:

https://hn.algolia.com/?query=Computational%20Complexity%20o...

[2] Logic, Optimization, and Constraint Programming: A Fruitful Collaboration - John Hooker - CMU (2023) [video]:

https://www.youtube.com/live/TknN8fCQvRk

[3] Google OR-Tools:

https://developers.google.com/optimization

[4] MiniZinc:

https://www.minizinc.org/