Computational complexity theory, as a branch of the theory of computation in computer science, investigates the problems related to the amounts of resources required for the execution of algorithms (e.g., execution time), and the inherent difficulty in providing efficient algorithms for specific computational problems.
A typical question of the theory is, "As the size of the input to an algorithm increases, how do the running time and memory requirements of the algorithm change and what are the implications and ramifications of that change?" In other words, the theory, among other things, investigates the scalability of computational problems and algorithms. In particular, the theory places practical limits on what computers can accomplish.
An important aspect of the theory is to categorize computational problems and algorithms into complexity classes. The most important open question of complexity theory is whether the complexity class P is the same as the complexity class NP, or is a strict subset as is generally believed. Shortly after the question was first posed, it was realised that many important industry problems in the field of operations research are of an NP subclass called NP-complete. NP-complete problems have the property that solutions to these problems are quick to check, yet the current methods to find solutions are not "efficiently scalable" (as described more formally below). More importantly, if the NP class is larger than P, then no efficiently scalable solutions exist for these problems.
The openness of the P-NP problem prompts and justifies various research areas in the computational complexity theory, such as identification of efficiently solvable special cases of common computational problems, study of the computational complexity of finding approximate or heuristic solutions, as well as research into the hierarchies of complexity classes.
Reference:
http://en.wikipedia.org/wiki/Computational_complexity_theory
