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

Problems and Reductions
can be made clear by an example. The traveling salesman problem is defined thus:
Input: A weighted graph G.
Output: Which tour minimizes ?
Thus any weighted graph defines an instance of TSP. Each particular instance has at least one minimum cost tour. The
general traveling salesman problem asks for an algorithm to find the optimal tour for all possible instances.
Any problem with answers restricted to yes and no is called a decision problem. Most interesting optimization problems can
be phrased as decision problems that capture the essence of the computation. For example, the traveling salesman decision
problem can be defined thus:
Input: A weighted graph G and integer k.
Output: Does there exist a TSP tour with cost ? It should be clear that the decision version captures the heart of the
traveling salesman problem, for if you had a program that gave fast solutions to the decision problem, you could do a binary
search with different values of k to quickly hone in on the correct solution.
Therefore, from now on we will talk only about decision problems, because it is easier to reduce one problem to another
when the only possible answers to both are true or false.
Next: Simple Reductions Up: Intractable Problems and Approximations Previous: Intractable Problems and
Approximations
Algorithms
Mon Jun 2 23:33:50 EDT 1997
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