# Are there NP COMPLETE problems that are “easy” in practice?

NP COMPLETE problems are hard in the worst case (assuming $$P \neq NP$$). What that means is that for every polynomial $$p$$, sufficiently large integer $$n$$, and algorithm $$A$$, there is an instance $$x$$ of size $$n$$ for which the algorithm takes more than $$p(n)$$ time. But this is one instance for every (sufficiently large) $$n$$. In principle, this could be the only hard instance for that value of $$n$$, and all other instances for that $$n$$ could be easy. So how hard are NP COMPLETE problems in practice?

I'll refine that question to: Are there NP COMPLETE problems that are somehow easy in practice?

The definition of "easy" is left open. One definition may be given by average-case complexity, another one could be given "smooth complexity", another could be given by Fixed Parameter Tractability, efficient approximability, polynomial-time solvability with an advice oracle, efficiency in practice without a mathematical definition etc. I'm hoping that by leaving the definition of "easy" open, I can get a wider range of answers. Any definition of "easy" should imply that the problem is easy "in real life" or "in practice". Also, don't assume I know any of that stuff I just listed in any detail.

• You already seem to know that not every instance of an NPC problem is hard. So what's the question, really? SAT has been heavily studied and you will have no problem finding "real instances" that are easy in practice. – Juho Feb 3 at 0:23