And so far, for the hard problems, the solutions have required on the order of formula_6 computations. For any particular problem, people have found ways to reduce the number of computations needed. One might figure out that a way to do just 1% of the worst-case number of computation and that saves a "lot" of computing, but that is still formula_9 computations. And every extra rock still doubles the number of computations needed to solve the problem. There are insights that can produce methods to do even fewer computations producing variations of the model: e.g. formula_10. But the exponential function still dominates as formula_5 grows.
Consider the problem of scheduling exams (described above). But suppose, next, that there are 15000 students. There's a computer program that takes the schedules of all 15000 students. It runs in an hour and outputs an exam schedule so that all students can do their exams in one week. It satisfies lots of rules (no back-to-back exams, no more than 2 exams in any 28 hour period, ...) to limit the stress of exam week. The program runs for one hour at mid-term break and everyone knows his/her exam schedule with plenty of time to prepare.
The next year, though, there are 10 more students. If the same program runs on the same computer then that one hour is going to turn into formula_12 hours, because every additional student doubles the computations. That's formula_13 weeks! If there were 20 more students, then
Thus, for formula_18 students, it takes one hour. For formula_19 students, it takes formula_17 years.
As you can see, exponential functions grow really fast. Most mathematicians believe that the hardest NP problems require exponential time to solve.
Mathematicians can show that there are some NP problems that are NP-Complete. An NP-Complete problem is at least as difficult to solve as any other NP problem. This means that if someone found a method to solve any NP-Complete problem quickly, they could use that same method to solve "every" NP problem quickly. All of the problems listed above are NP-Complete, so if the salesman found a way to plan his trip quickly, he could tell the teacher, and she could use that same method to schedule the exams. The farmer could use the same method to determine how many boxes she needs, and the woman could use the same method to find a way to build her towers.
Because a method that quickly solves one of these problems can solve them all, there are many people who want to find one. However, because there are so many different NP-Complete problems and nobody so far has found a way to solve even one of them quickly, most experts believe that solving NP-Complete problems quickly is not possible.
In computational complexity theory, the complexity class NP-complete (abbreviated NP-C or NPC), is a class of problems having two properties:
NP-complete is a subset of NP, the set of all decision problems whose solutions can be verified in polynomial time; NP may be equivalently defined as the set of decision problems solved in polynomial time on a machine. A problem p in NP is also in NPC if and only if every other problem in NP is transformed into p in polynomial time. NP-complete was to be used as an adjective: problems in the class NP-complete were as NP+complete problems.
NP-complete problems are studied because the ability to quickly verify solutions to a problem (NP) seems to correlate with the ability to quickly solve problem (P). It is found every problem in NP is quickly solved—as called the P = NP: problem set. The single problem in NP-complete is solved quickly, faster than every problem in NP also quickly solved, because the definition of an NP-complete problem states every problem in NP must be quickly reducible to every problem in NP-complete (it is reduced in polynomial time).
The Boolean satisfiability problem is known to be NP complete. In 1972, Richard Karp formulated 21 problems that are known to be NP-complete. These are known as Karp's 21 NP-complete problems. They include problems such as the Integer programming problem, which applies linear programming techniques to the integers, the knapsack problem, or the vertex cover problem.
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Comprehensive list of the recordings made.
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