Travelling salesman very rough min and max estimates
It is conjectured (but unproven) that the Held-Karp relaxation (not the Held-Karp algorithm) has an integrality ratio of 4/3 (on metric TSP). Various proofs exists for even better bounds if some assumptions are made.

Practically it performs even better, gaps of ~2% are normal, and there is a lot of literature about adding more cuts that get that gap (typically) a fair way below 1%, but that would not fit with your goal of "fast and easy to implement" anymore. Just the basic degree constraints + lazy sub-tour elimination is, when using an existing LP solver, both fast and easy, and a lot better than taking one of the well known upper bounds and dividing them by their approximation ratio (which tends to be give terrible results, because they typically approximate much better than their worst case, resulting in an overly optimistic lower bound).