A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Other complexity measures are used, such as the amount of communication, the number of gates in a circuit. One of the roles of computational complexity theory is to determine the limits on what computers can. Closely related fields in computer science are analysis of algorithms. More precisely, computational complexity theory tries to classify problems that can or cannot be solved with appropriately restricted resources, a computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input string for a problem is referred to as a problem instance. In computational complexity theory, a problem refers to the question to be solved. In contrast, an instance of this problem is a rather concrete utterance, for example, consider the problem of primality testing. The instance is a number and the solution is yes if the number is prime, stated another way, the instance is a particular input to the problem, and the solution is the output corresponding to the given input. For this reason, complexity theory addresses computational problems and not particular problem instances, when considering computational problems, a problem instance is a string over an alphabet. Usually, the alphabet is taken to be the binary alphabet, as in a real-world computer, mathematical objects other than bitstrings must be suitably encoded. For example, integers can be represented in binary notation, and graphs can be encoded directly via their adjacency matrices and this can be achieved by ensuring that different representations can be transformed into each other efficiently. Decision problems are one of the objects of study in computational complexity theory. A decision problem is a type of computational problem whose answer is either yes or no. A decision problem can be viewed as a language, where the members of the language are instances whose output is yes. The objective is to decide, with the aid of an algorithm, if the algorithm deciding this problem returns the answer yes, the algorithm is said to accept the input string, otherwise it is said to reject the input. An example of a problem is the following
A traveling salesman tour through Germany's 15 largest cities.
A decision problem has only two possible outputs, yes or no (or alternately 1 or 0) on any input.
An illustration of a Turing machine
A representation of the relation among complexity classes