Just a thought, just because a general problem is NPHard doesn't mean that we can't find specific solutions quickly or that a given input is hard to search for. If the downstream effect results in an order of magnitude less work, it makes sense, it's just a tradeoff.
The one I liked was the postgres genetic optimizer. https://www.postgresql.org/docs/17/geqo-pg-intro.html The theory being an exhaustive search of all possible query plans is np-hard and would take too long, so you do a limited, iterative, best fit search. My understanding is it never worked super great and would only be used if your query exceeded some high level of complexity. I distinctly remember it being removed…
Beam search approximates NP-hard solutions (make wider beam, have better approximation), is very old and it is used in SQLite query planner: https://www.sqlite.org/queryplanner-ng.html