Travelling salesman very rough min and max estimates Is there a way to find very rough minimum and maximum estimates for the travelling salesman problem? The estimates only need to be within the roughly same magnitude, but it's important that the minimum estimate is lower (or equal) to the actual distance and that the maximum estimate is higher (or equal) to the actual distance. 

EDIT: 
Metric TSP unfortunately does not apply.
To add some context: I'm searching for non-dominated fronts (pareto fronts) for a Multiobjective TSP, e.g. optimizing for multiple variables like travel time and distance. I'm using NSGA-II which requires an upper and lower bound for each objective function, which translates into rough estimates for single-objective TSP, with the constraint that the upper estimate has to be higher than the real max distance and vice versa.
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).