  As discussed, natural problems tend to be complete for important complexity classes, namely the ones in the diagram and only a very few others.  This fascinating phenomenon means that algorithms and complexity are more than abstract concepts; they are important at a practical level.  We have had remarkable success in proving that our problem of interest is complete for a well-known complexity class.  If the class is contained in P, then we can usually just look up a known efficient algorithm.  Otherwise, we must look at simplifications or approximations of our problem which may be feasible.