I normally do machine learning work, and when I'm evaluating an algorithm on a data set, I always use cross-validation to determine how effective the algorithm is. Is there a similar method for evaluating approximation algorithms on specific instances of NP-hard problems? If not, how do you determine which algorithm is best for a given problem domain in practice?
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Crossvalidation is used to test the difference in behaviour between training and test data. There's no equivalent notion for approximations. However, for many problems it's possible (often via an LP relaxation) to get a good lower bound for a problem. This can be used to test the quality of a given approximation algorithm in practice.
For a good example of this, you should look at David Johnson's survey of experimental work on the travelling salesman problem.