Thread: Is this NP-complete

Hello,

If there is a decision problem, which is NP-complete,
question: can the problem A be done in k steps?

then, is the following problem also np-complete
question: can the problem A be done in k+c steps, where c is some fixed constant?

Basically, I would like to know if a problem is NP-complete, then whether finding a solution which is a constant away from the optimal is also NP-complete or not?

Thanks

You're asking two different questions there.

The decision problem "can A be done in k steps" doesn't look for an actual solution, and if you knew (k+c) and c, you'd know k, so for a fixed c, yes, it'd still be NP complete.

The question of "find a solution that is within some error of optimal" is not a decision problem, so it can't be NP complete no matter what error you use. Such problems can be in P though as there are numerous algorithms that give such answers to otherwise difficult questions.