Small generating set-finding problem for permutation encodings

Description
A group $$G$$ acting on a set $$S$$ of size $$n$$, described by means of a generating set $$A$$ of $$G$$.

Find a generating set for $$G$$ whose size is bounded by some fixed polynomial in $$n$$, in time proportional to $$|A|$$ times a fixed polynomial in $$n$$.

Particular algorithms

 * Jerrum's filter outputs a Jerrum-reduced generating set, which has size at most $$n-1$$. The time it takes is $$|A|n^4$$. The critical step in this algorithm in the iterated step of making changes at cycles.
 * Sims filter outputs a Sims-reduced generating set.

Reduction to generating set-checking problem
The generating set-finding problem has a nondeterministic reduction to the generating set-checking problem: first guess a generating set, then check if it works.

Importance
Many problems, including the testing of group properties and subgroup properties, have algorithms that proceed much faster if a generating set of small size is available. Thus, the small generating set-finding algorithms are typically used in conjunction with the generating set-dependent algorithms.