Computational complexity theory
Complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps it takes to solve a problem) and space (how much memory it takes). Other resources can also be considered, such as how many parallel processors are needed to solve a problem in parallel. Complexity theory differs from computability theory, which deals with whether a problem can be solved at all, regardless of the resources required.
After the theory explaining which problems can be solved and which cannot be, it was natural to ask about the relative computational difficulty of computable functions. This is the subject matter of computational complexity.
A single "problem" is an entire set of related questions, where each question is a finite-length string. For example, the problem FACTORIZE is: given an integer written in binary, return all of the prime factors of that number. A particular question is called an instance. For example, "give the factors of the number 15" is one instance of the FACTORIZE problem.
The time complexity of a problem is the number of steps that it takes to solve an instance of the problem as a function of the size of the input (usually measured in bits), using the most efficient algorithm. To understand this intuitively, consider the example of an instance that is n bits long that can be solved in n² steps. In this example we say the problem has a time complexity of n². Of course, the exact number of steps will depend on exactly what machine or language is being used. To avoid that problem, we generally use Big O notation. If a problem has time complexity O(n²) on one typical computer, then it will also have complexity O(n²) on most other computers, so this notation allows us to generalize away from the details of a particular computer.
Example: Mowing grass has linear complexity because it takes double the time to mow double the area. However, looking up something in a dictionary has only logarithmic complexity because a double sized dictionary only has to be opened one time more (e.g. exactly in the middle - then the problem is reduced to the half).
Much of complexity theory deals with decision problems. A decision problem is a problem where the answer is always YES/NO. For example, the problem IS-PRIME is: given an integer written in binary, return whether it is a prime number or not. A decision problem is equivalent to a language, which is a set of finite-length strings. For a given decision problem, the equivalent language is the set of all strings for which the answer is YES.
Decision problems are often considered because an arbitrary problem can always be reduced to a decision problem. For example, the problem HAS-FACTOR is: given integers n and k written in binary, return whether n has any prime factors less than k. If we can solve HAS-FACTOR with a certain amount of resources, then we can use that solution to solve FACTORIZE without much more resources. Just do a binary search on k until you find the smallest factor of n. Then divide out that factor, and repeat until you find all the factors.
Complexity theory often makes a distinction between YES answers and NO answers. For example, the set NP is defined as the set of problems where the YES instances can be checked "quickly" (i.e. in polynomial time). The set Co-NP is the set of problems where the NO instances can be checked quickly. The "Co" in the name stands for "complement". The complement of a problem is one where all the YES and NO answers are swapped, such as IS-COMPOSITE for IS-PRIME.
An important result in complexity theory is the fact that no matter how hard a problem can get (i.e. how much time and space resources it requires), there will always be even harder problems. At least for time complexity, and for polynomial-time decision problems, this is determined by the time hierarchy theorem. A similar space hierarchy theorem can also be derived.
The complexity class P is the set of decision problems that can be solved by a deterministic machine in polynomial time. This class corresponds to an intuitive idea of the problems which can be effectively solved in the worst cases.
The complexity class NP is the set of decision problems that can be solved by a non-deterministic machine in polynomial time. This class contains many problems that people would like to be able to solve effectively, including the Boolean satisfiability problem, the Hamiltonian path problem and the Vertex cover problem. All the problems in this class have the property that their solutions can be checked effectively.
The P = NP question
The question of whether P is the same set as NP is the most important open question in theoretical computer science. There is even a $1,000,000 prize for solving it. (See complexity classes P and NP and oracles).
Questions like this motivate the concepts of hard and complete. A set of problems X is hard for a set of problems Y if every problem in Y can be transformed easily into some problem in X with the same answer. The definition of "easily" is different in different contexts. The most important hard set is NP-hard. Set X is complete for Y if it is hard for Y, and is also a subset of Y. The most important complete set is NP-complete. See the articles on those two sets for more detail on the definition of "hard" and "complete".
Problems that are solvable in theory, but can't be solved in practice, are called intractable. What can be solved "in practice" is open to debate, but in general only problems that have polynomial-time solutions are solvable for more than the smallest inputs. Problems that are known to be intractable include those that are EXPTIME-complete. If NP is not the same as P, then the NP-complete problems are also intractable.
To see why exponential-time solutions are not usable in practice, consider a problem that requires 2n operations to solve (n is the size of the input). For a relatively small input size of n=100, and assuming a computer that can perform 1010 (10 giga) operations per second, a solution would take about 4*1012 years, much longer than the current age of the universe.
- Manindra Agrawal
- Sanjeev Arora
- Laszlo Babai
- Manuel Blum, who developed an axiomatic complexity theory based on his Blum axioms
- Allan Borodin
- Stephen Cook
- Lance Fortnow
- Juris Hartmanis
- Russell Impagliazzo
- Richard Karp
- Marek Karpinski
- Leonid Levin
- Richard Lipton
- Noam Nisan
- Christos H. Papadimitriou
- Alexander Razborov
- Walter Savitch
- Michael Sipser
- Richard Stearns
- Madhu Sudan
- Leslie Valiant
- Umesh Vazirani
- Avi Wigderson
- Andrew Yao
- Eugene Yarovoi
- Borja Sotomayor
- List of important publications in computational complexity theory
- List of open problems in computational complexity theory
- List of computability and complexity topics
- L. Fortnow, Steve Homer (2002/2003). A Short History of Computational Complexity. In D. van Dalen, J. Dawson, and A. Kanamori, editors, The History of Mathematical Logic. North-Holland, Amsterdam.
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