Approximating the mean speedup in trace monoids

Traces can be viewed as parallel processes and the "mean speedup" of a trace monoid has been introduced as a measure of "intrinsic parallelism" of the monoid. We study the problem of computing the mean speedup under two conditions: uniform distribution on the words of given length and uniform distribution on the traces of given height. In the first case, we give an approximability result showing a probabilistic fully polynomial time approximation scheme, while, in the second case, we prove that the problem is NP-hard to approximate within n1-c for every ∊ > 0, unless NP = coR. A further consequence is the hardness of the problem of generating traces of a given height uniformly at random.