Chapter 7: Mechanisms in Programs and Nature

Section 8: The Problem of Satisfying Constraints

NP completeness [and constraint satisfaction]

Finding 2D patterns that satisfy the constraints in the previous section is in general a so-called NP-complete problem. And this means that no known algorithm can be expected to solve this problem exactly for a size n array (say with given boundaries) in much less than 2^n steps (see page 1145). The same is true even if one allows a small fraction of squares to violate the constraints. However, the 1D version of the problem is not NP-complete, and in fact there is a specific rather efficient algorithm described on page 954 for solving it. Nevertheless, the procedures discussed in this section do not manage to make use of such specific algorithms, and in fact typically show little difference between problems that are and are not formally NP-complete.

From Stephen Wolfram: A New Kind of Science [citation]