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Even if we accept that P = NP, what about the speci c problems on the left side of the table On the basis of what evidence do we believe that these particular problems have no ef cient algorithm (besides, of course, the historical fact that many clever mathematicians and computer scientists have tried hard and failed to nd any) Such evidence is provided by reductions, which translate one search problem into another What they demonstrate is that the problems on the left side of the table are all, in some sense, exactly the same problem, except that they are stated in different languages What s more, we will also use reductions to show that these problems are the hardest search problems in NP if even one of them has a polynomial time algorithm, then every problem in NP has a polynomial time algorithm Thus if we believe that P = NP, then all these search problems are hard We de ned reductions in 7 and saw many examples of them Let s now specialize this de nition to search problems A reduction from search problem A to search problem B is a polynomial-time algorithm f that transforms any instance I of A into an instance f (I) of B, together with another polynomial-time algorithm h that maps any solution S of f (I) back into a solution h(S) of I; see the following diagram If f (I) has no solution, then neither does 240
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I These two translation procedures f and h imply that any algorithm for B can be converted into an algorithm for A by bracketing it between f and h
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And now we can nally de ne the class of the hardest search problems A search problem is NP-complete if all other search problems reduce to it This is a very strong requirement indeed For a problem to be NP-complete, it must be useful in solving every search problem in the world! It is remarkable that such problems exist But they do, and the rst column of the table we saw earlier is lled with the most famous examples In Section 83 we shall see how all these problems reduce to one another, and also why all other search problems reduce to them
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The two ways to use reductions
So far in this book the purpose of a reduction from a problem A to a problem B has been straightforward and honorable: We know how to solve B ef ciently, and we want to use this knowledge to solve A In this chapter, however, reductions from A to B serve a somewhat perverse goal: we know A is hard, and we use the reduction to prove that B is hard as well! If we denote a reduction from A to B by A B then we can say that dif culty ows in the direction of the arrow, while ef cient algorithms move in the opposite direction It is through this propagation of dif culty that we know NP-complete problems are hard: all other search problems reduce to them, and thus each NP-complete problem contains the complexity of all search problems If even one NP-complete problem is in P, then P = NP Reductions also have the convenient property that they compose If A B and B C, then A C To see this, observe rst of all that any reduction is completely speci ed by the pre- and postprocessing functions f and h (see the reduction diagram) If (f AB , hAB ) and (fBC , hBC ) de ne the reductions from A to B and from B to C, respectively, then a reduction from A to C is given by compositions of these functions: f BC fAB maps an instance of A to an instance of C and hAB hBC sends a solution of C back to a solution of A This means that once we know a problem A is NP-complete, we can use it to prove that a new search problem B is also NP-complete, simply by reducing A to B Such a reduction establishes that all problems in NP reduce to B, via A 241
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