Difference between Problems and Instances

The difference between the two concepts viz., ‘problem’ and ‘instance’, can be understood in terms of the following example. An instance of a problem is also called a question. We know that the roots of a general quadratic equation

 (1)

are given by the equation

 (2)

where a, b, c may be any real numbers except the restriction that a ≠ 0.

Now, if we take a = 3, b = 4 and c = 1, we get the particular equation

 (3)

Using (2), the roots of (3) are given by

With reference to the above discussion, the issue of finding roots of the general quadratic equation ax2 + bx + c = 0, with a ≠ 0 is called a problem, whereas the issue
of finding the roots of the particular equation

is called a question or an instance of the (general) problem.

In general, a problem may have a large, possibly infinite, number of instances. The above-mentioned problem of finding the roots of the quadratic equation

with a ≠ 0, b and c as real numbers, has infinitely many instances, each obtained by giving some specific real values to a, b and c, taking care that the value assigned to a is not zero. However, all problems may not be of generic nature. For some problems, there may be only one instance/question corresponding to each of the problems. For example, the problem of finding out the largest integer that can be stored or can be arithmetically operated on, in a given computer, is a single-instance problem. Many of the interesting problems like the ones given below, are just single-instance problems.