Reducibility and Completeness

There is little doubt that the notion of reducibility is the most useful tool that complexity theory has delivered to the rest of the computer science community. For most computational problems that arise in real-world applications, such as the Traveling Salesperson Problem, we still know little about their deterministic time or space complexity. We cannot now tell whether classes such as P and NP are distinct. And yet, even without such hard knowledge, it has been useful in practice to take some new problem A whose complexity needs to be analyzed, and announce that A has roughly the same complexity as the Traveling Salesperson Problem, by exhibiting efficient ways of reducing each problem to the other. Thus we can say a lot about problems being equivalent in complexity to each other, even if we cannot pinpoint what that complexity is. One reason for this success is that when one partitions the many thousands of real-world computational problems into equivalence classes according to the reducibility relation, there are surprisingly few classes in this partition. Thus the complexity of almost any problem arising in practice can be classified by showing that it is equivalent to one of a short list of representative problems. It was not originally expected that this would be the case. Even more amazingly, these “representative problems” correspond in a natural way to abstract models of computation—that is, they correspond to complexity classes. These classes were defined in Chapter 22 using a small set of abstract machine concepts: Turing machines, nondeterminism, alternation, time, space, circuits. With this and a few simple functions that define time and space bounds, we are able to characterize the complexity of the overwhelming majority of natural computational problems—most of which bear no topical resemblance to any question about Turing machines. This tool has been much more successful than we had any right to expect it would be. All this leads us to believe that it is no mere accident that problems easily lend themselves to being placed in one class or another. That is, we are disposed to think that these classes really are distinct, that the classification is real , and that the mathematics developed to deal with them really does describe some important aspect of nature. Nondeterministic Turing machines, with their magic ability to soar through immense search spaces, seem to be much more powerful than our mundane deterministic machines, and this reinforces our belief. However, until P vs. NP and similar long-standing questions of complexity theory are completely resolved, our best method of understanding the complexity of real-world problems is to use the classification provided by reducibility, and to trust in a few plausible conjectures.