Types of Computational Problems
A decision problem is a computational problem where the answer for every instance is either yes or no. An example of a decision problem is primality testing:
- "Given a positive integer n, determine if n is prime."
A decision problem is typically represented as the set of all instances for which the answer is yes. For example, primality testing can be represented as the infinite set
- L = {2, 3, 5, 7, 11, ...}
In a search problem, the answers can be arbitrary strings. For example, factoring is a search problem where the instances are (string representations of) positive integers and the solutions are (string representations of) collections of primes.
A search problem is represented as a relation over consisting of all the instance-solution pairs, called a search relation. For example, primality can be represented as the relation
- R = {(4, 2), (6, 2), (6, 3), (8, 2), (9, 3), (10, 2), (10, 5)...}
which consist of all pairs of numbers (n, p), where p is a nontrivial prime factor of n.
A counting problem asks for the number of solutions to a given search problem. For example, the counting problem associated with primality is
- "Given a positive integer n, count the number of nontrivial prime factors of n."
A counting problem can be represented by a function f from {0, 1}* to the nonnegative integers. For a search relation R, the counting problem associated to R is the function
- fR(x) = |{y: (x, y) ∈ R}|.
An optimization problem asks for finding the "best possible" solution among the set of all possible solutions to a search problem. One example is the maximum independent set problem:
- "Given a graph G, find an independent set of G of maximum size."
Optimization problems can be represented by their search relations.
Read more about this topic: Computational Problem
Famous quotes containing the words types of, types and/or problems:
“... there are two types of happiness and I have chosen that of the murderers. For I am happy. There was a time when I thought I had reached the limit of distress. Beyond that limit, there is a sterile and magnificent happiness.”
—Albert Camus (1913–1960)
“Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly.”
—Talcott Parsons (1902–1979)
“It is not impossible, of course, after such an administration as Roosevelt’s and after the change in method that I could not but adapt in view of my different way of looking at things, that questions should arise as to whether I should go back on the principles of the Roosevelt administration.... I have a government of limited power under a Constitution, and we have got to work out our problems on the basis of law. Now, if that is reactionary, then I am a reactionary.”
—William Howard Taft (1857–1930)