Algorithm to compose identity from a set of permutations

Given a subset P of all the possible permutations of a fixed set of elements, is there a non-exponential or optimized algorithm for computing the smallest composition of P that yields the identity permutation?

Assuming that "smallest composition" means smallest number of permutations used in the composition, then the NP-complete Pancake Flipping Problem is a special case of your problem.

StackExchange Computer Science Q#43084, answer score: 4