The.Algorithm.Design.Manual.Springer-Verlag.1998

Problems and Reductions
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Approximations
Problems and Reductions
Throughout this book we have encountered problems, such as the traveling salesman problem, for which we couldn't find
any efficient algorithm. By the early 1970s, literally hundreds of problems were stuck in this swamp. The theory of NPcompleteness
provided the tools needed to show that all of these problems were really the same thing.
The key idea behind demonstrating the hardness of a problem is that of a reduction. Suppose that I gave you the following
algorithm to solve the Bandersnatch problem:
Bandersnatch(G)
Translate the input G to an instance of the Bo-billy problem Y.
Call the subroutine Bo-billy on Y to solve this instance.
Return the answer of Bo-billy(Y) as the answer to Bandersnatch(G).
It is important to see that this algorithm correctly solves the Bandersnatch problem provided that the translation to Bo-billy
always preserves the correctness of the answer. In other words, the translation has the property that for any instance of G,
Bandersnatch(G) = Bo-billy(Y). A translation of instances from one type of problem to instances of another type such that
the answers are preserved is called a reduction.
Now suppose this reduction translates G to Y in O(P(n)) time. There are two possible implications:
● If my Bo-billy subroutine ran in O(P'(n)), this means I could solve the Bandersnatch problem in O(P(n)+P'(n)) by
spending the time to translate the problem and then the time to execute the Bo-Billy subroutine.
● If I know that is a lower bound on computing Bandersnatch, meaning there definitely exists no faster way to
solve it, then must be a lower bound to compute Bo-billy. Why? If I could solve Bo-billy any faster,
then I could solve Bandersnatch in faster time by using the above simulation, thus violating my lower bound. This
implies that there can be no way to solve Bo-billy any faster than claimed.
This second argument is the approach that we will use to prove problems hard. Essentially, this reduction shows that Bobilly
is at least as hard as Bandersnatch, and therefore once we believe that Bandersnatch is hard, we have a tool for proving
other problems hard.
Reductions, then, are operations that convert one problem into another. To describe them, we must be somewhat rigorous in
our definition of a problem. A problem is a general question, with parameters for the input and conditions on what
constitutes a satisfactory answer or solution. An instance is a problem with the input parameters specified. The difference
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