ALRch28.pdf - Reducibility and Completeness Chapter 28 of the forthcoming CRC Handbook on Algorithms and Theory of Computation Eric Allender1 Rutgers

ALRch28.pdf - Reducibility and Completeness Chapter 28 of...

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Reducibility and CompletenessChapter 28 of the forthcomingCRC Handbook on Algorithms and Theory of ComputationEric Allender1Rutgers UniversityMichael C. Loui2University of Illinois at Urbana-ChampaignKenneth W. Regan3State University of New York at Buffalo1IntroductionThere is little doubt that the notion ofreducibilityis the most useful tool that complexity theoryhas delivered to the rest of the computer science community.For most computational problems that arise in real-world applications, such as the TravelingSalesperson Problem, we still know little about their deterministic time or space complexity. Wecannot now tell whether classes such asPandNPare distinct. And yet, even without such hardknowledge, it has been useful in practice to take some new problemAwhose complexity needsto be analyzed, and announce thatAhas roughly the same complexity as Traveling Salesperson,by exhibiting efficient ways of reducing each problem to the other. Thus we can say a lot aboutproblems being equivalent in complexity to each other, even if we cannot pinpoint what thatcomplexity is.One reason this has succeeded is that, when one partitions the many thousands of real-worldcomputational problems into equivalence classes according to the reducibility relation, there is asurprisingly small number of classes of this partition. Thus, the complexity of almost any prob-lem arising in practice can be classified by showing that it is equivalent to one of a short list ofrepresentative problems. It was not originally expected that this would be the case.1Supported by the National Science Foundation under Grant CCR-9509603. Portions of this work were performedwhile a visiting scholar at the Institute of Mathematical Sciences, Madras, India.2Supported by the National Science Foundation under Grant CCR-9315696.3Supported by the National Science Foundation under Grant CCR-9409104.1
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Even more amazingly, these “representative problems” correspond in a very natural way toabstract models of computation—that is, they correspond to complexity classes.These classeswere defined in the last chapter using a small set of abstract machine concepts: Turing machines,nondeterminism, alternation, time, space. With this and a few simple functions that define time andspace bounds, we are able to characterize the complexity of the overwhelming majority of naturalcomputational problems—most of which bear no topical resemblance to any question about Turingmachines. 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 tobeing placed in one class or another. That is, we are disposed to think that these classes reallyaredistinct, that the classification isreal, and that the mathematics developed to deal with themreally does describe some important aspect of nature.Nondeterministic Turing machines, withtheir ability to magically soar through immense search spaces,seem
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