P: The problems that can be solved in polynomial time.
NP: The problems whose solutions can be verified in polynomial time.
NP-complete: The problems in the set of NP problems and also in the
set of NP-hard problems.Although any given solution to
such a problem can be verified quickly, there is no known
efficient way to locate a solution in the first place;
indeed, the most notable characteristic of NP-complete
problems is that no fast solution to them is known.
What makes NP-complete problems important is that if a
deterministic polynomial time algorithm can be found to
solve one of them, every NP problem is solvable in
polynomial time (one problem to rule them all).--Cook's theorem.
NP-hard: The problems that are at least as hard as NP-complete problems.
NP-hard problems do not have to be in NP.
Usually there are three kinds of algorithms to solve NP-complete and
NP-hard problems:
1. Backtracking algorithms with good prunning techniques.
2. Heuristic algorithms.
3. Approximation algorithms.
Differences between heuristic algorithms and approximation algorithms are
approximation algorithms are like putting money in the banks -- you always
get 3% benefit a year (approximation algorithms always return solutions
with a guarantee attached, you can never go too far wrong when using them)
Heuristic algorithms are like putting your money into the stock market,
you may do much better than approximation algorithms or even get the
optimal solution, but they are without guarranty.
