TITLE: How slow are direct solutions of NP-complete problems on computers?
QUESTION [7 upvotes]: Sometimes I see that people call a problem NP-hard and because of that refuse to create computer algorithms that directly solve it. I think I've never read actual benchmark results for such problems. Are there any? Are there any NP-complete problems with not very bad performance?

REPLY [6 votes]: For Boolean Satisfiability (SAT), there is Sat Competition: http://www.satcompetition.org/
State of the art SAT solvers often perform very efficiently, far faster than exponential. At the above link there should be results to see how large instances were solved.
Sucess story: Using SAT solver, important partial results about a conjecture of Erdos were found. The formula was quite big, for details see here: http://cgi.csc.liv.ac.uk/~konev/SAT14/
Sometimes heuristic algorithms perform very fast on graph problems.
For SAT, if you generate random SAT instance, very likely state of the art SAT solver will solve it very fast. (There was very similar question about this here).