Computational complexity theory deals with the resources needed to solve problems on a computer system. Problems are classified by difficulty as polynomial (P) or non-polynomial (NP). Solving a computation requires resources such as time, storage space, and hardware. NP problems are more complex and may require more advanced options. Classifying P versus NP problems can be complex, and some problems do not fit neatly into either category. The process of exploring programs and developing approaches to solving them is an important area of mathematics and computer science.
Computational complexity theory is an area of mathematics and computer science that deals with the resources needed to solve problems on a computer system. A number of techniques are available for determining the resource requirements of a problem. Some problems may not be feasible on existing computer systems due to their resource demands. Researchers classify problems by difficulty and can divide calculations into polynomial (P) and non-polynomial (NP) problems.
Solving a computation requires resources such as time, storage space, and hardware. A computer system may have limitations that make a problem functionally impossible to solve because it lacks the resources available. As computer technology improves, a previously unsolvable problem may become solvable with the help of new technologies and research in the field of computational complexity theory. The solvability of a problem is not necessarily determined by its complexity but by the algorithms used to solve it.
In computational complexity theory, a problem P is a problem that can be solved in polynomial time with a simple algorithm. It may still require substantial resources, but it is both fixable and computer controllable. Such problems might be thought to be fixable quickly provided a computer has the resources available to handle the necessary calculations.
NP problems are more complex. It is not possible to apply a single algorithm and it may be necessary to use more advanced options, such as parallel Turing machines which can explore several options. The problem may be fixable this way, but it will require substantially more resources. Such problems may be easier for human operators who are capable of advanced logical thinking, because the game-changer is often one of logic rather than pure computational difficulty. The traveling salesman problem, in which the goal is to find the most efficient route between a number of cities along a route, is a classic example of an NP problem in computational complexity theory.
Classifying P versus NP problems through computational complexity theory can be a complex task, and problems may drift back and forth across the gap. A small set of computational problems do not fit neatly into either category and are sometimes classified as neither to reflect this. It may eventually be possible to develop an algorithm for solving an NP problem and, in some cases, it may apply to other problems that have a similar structure. In others, however, it may be specific to the problem. The process of exploring such programs and developing approaches to solving them is an important area of mathematics and computer science that contributes to the development of high-power and advanced computing systems.
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