Complexity Theory

Computational complexity is the study of the difficulty of solving computational problems, in terms of the required computational resources, such as time and space (memory). Whereas the analysis of algorithms focuses on the time or space requirements of an individual algorithm for a specific problem (such as sorting), complexity theory focuses on the complexity class of problems solvable in the same amount of time or space. Most common computational problems fall into a small number of complexity classes. Two important complexity classes are P, the set of problems that can be solved in polynomial time, and NP, the set of problems whose solutions can be verified in polynomial time. By quantifying the resources required to solve a problem, complexity theory has profoundly affected our thinking about computation. Computability theory establishes the existence of undecidable problems, which cannot be solved in principle regardless of the amount of time invested. However, computability theory fails to find meaningful distinctions among decidable problems. In contrast, complexity theory establishes the existence of decidable problems that, although solvable in principle, cannot be solved in practice, because the time and space required would be larger than the age and size of the known universe [Stockmeyer and Chandra, 1979]. Thus, complexity theory characterizes the computationally feasible problems. The quest for the boundaries of the set of feasible problems has led to the most important unsolved question in all of computer science: Is P different from NP? Hundreds of fundamental problems, including many ubiquitous optimization problems of operations research, are NP-complete; they are the hardest problems in NP. If someone could find a polynomial-time algorithm for any one NP-complete problem, then there would be polynomial-time algorithms for all of them. Despite the concerted efforts of many scientists over several decades, no polynomial-time algorithm has been found for any NP-complete problem. Although we do not yet know whether P is different from NP, showing that a problem is NP-complete provides strong evidence that the problem is computationally infeasible and justifies the use of heuristics for solving the problem. In this chapter, we define P, NP, and related complexity classes. We illustrate the use of diagonalization and padding techniques to prove relationships between classes. Next, we define NP-completeness, and we show how to prove that a problem is NP-complete. Finally, we define complexity classes for probabilistic and interactive computations. Throughout this chapter, all numeric functions take integer arguments and produce integer values. All logarithms are taken to base 2. In particular, log n means dlog2 ne. Supported by the National Science Foundation under Grants CCF-0832787 and CCF-1064785. Supported by the National Science Foundation under Grants IIS-0832843, CNS-0851957, and DUE-1044207. Supported by the National Science Foundation under Grant CCR-9821040.