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# An Overview of Complexity Theory

## Abstract

Computational complexity theory (Shortly: Complexity Theory) has been a central area of theoretical computer science since its early development in the mid-1960s. Its subsequent rapid development in the next three decades, has not only established it as a rich, exciting theory, but also has shown strong influence on many other related areas in computer science, mathematics, and operation research (Du and Ko 2000). However, the notions of algorithms and complexity are meaningful only when they are defined in terms of formal computation models (Du and Ko 2000).

Apparently, we need some models to base the foundation of complexity theory on them. In this chapter, we introduce only three basic models: deterministic turing machine (DTM), non-deterministic turing machine (NTM) and Oracle machine models. It should be noted there are also some other models (see Du and Ko 2000).

Using such models, allows us to separate the complexity notion from any physical machine. Hence, we can measure the time complexity of algorithms and hardness of problems independent from a specific machine which runs the algorithm(s). It should be noted that these are just abstract models; means, are defined mathematically (Sipser 1996).

The structure of this chapter is as follows. We first discuss why we actually need complexity theory. Then, we introduce three basic models of computation: DTM and NTM and Oracle model. Then we present a brief introduction about the concept of big *O* notation which is widely used in the complexity theory. In Sect. 2.5, the decision problems as a special form of problems are described. Following this section, the basic concepts of reduction are presented, which help us to make relationships between different classes of complexity and also provide a rich tool to identify the unknown complexity class of a new problem. Finally, we introduce the most popular classes of complexity: *P>
*, *NP*, *NP*-complete and *NP*-hard. In each class, also, some known problems are presented.

## Keywords

Polynomial Time Decision Problem Turing Machine Travel Salesman Problem Travel Salesman Problem## References

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