Question

Difference between optimization problem and decision problem

Answer 1

Decision problems vs. optimization problems

Many problems of interest are optimization problems, in which each feasible (i.e., "legal") solution has an associated value, and we wish to find the feasible solution with the best value. For example, in a problem that we call SHORTEST-PATH, we are given an undirected graph G and vertices u and v, and we wish to find the path from u to v that uses the fewest edges. (In other words, SHORTEST-PATH is the single-pair shortest-path problem in an unweighted, undirected graph.) NP-completeness applies directly not to optimization problems, however, but to decision problems, in which the answer is simply "yes" or "no" (or, more formally, "1" or "0").

Although showing that a problem is NP-complete confines us to the realm of decision problems, there is a convenient relationship between optimization problems and decision problems. We usually can cast a given optimization problem as a related decision problem by imposing a bound on the value to be optimized. For SHORTEST-PATH, for example, a related decision problem, which we call PATH, is whether, given a directed graph G, vertices u and v, and an integer k, a path exists from u to v consisting of at most k edges.

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