NP-complete problems
Exhaustive search is usually intractable.
Please remember here the related property of an algorithm is not its running time on some instance(s), but how fast the running time increases with the instance size. As a result, for a given problem instance, an "intractable" algorithm may actually cost less time than an "efficient" algorithm!
Roughly speaking, P is the set of problems whose solutions can be found in polynomial time; NP is the set of problems whose solutions can be verified in polynomial time. Since to find a solution is a special way to verify a solution, P is a subset of NP. Now the problem is whether the two sets are the same. Though the intuitive answer is "No", it has not been proved yet.
A problem belongs to set NP-Complete, or NPC, if and only if all NP problems can be reduced into it. To prove a problem is in NPC, first proving it is in NP, then proving a known NPC problem can be reduced into it. The textbook gives many examples on how to reduce one NPC problem into another.
Here is a list of well known NPC problems. No polynomial-time solution has been found for any of them, though there is no proof that such a solution is impossible.
Example: heuristic search
Example: case-by-case problem solving