  We consider the optimization variant of the realizability problem for Prompt
Linear Temporal Logic, an extension of Linear Temporal Logic (LTL) by the
prompt eventually operator whose scope is bounded by some parameter. In the
realizability optimization problem, one is interested in computing the minimal
such bound that allows to realize a given specification. It is known that this
problem is solvable in triply-exponential time, but not whether it can be done
in doubly-exponential time, i.e., whether it is just as hard as solving LTL
realizability.
  We take a step towards resolving this problem by showing that the optimum can
be approximated within a factor of two in doubly-exponential time. Also, we
report on a proof-of-concept implementation of the algorithm based on bounded
LTL synthesis, which computes the smallest implementation of a given
specification. In our experiments, we observe a tradeoff between the size of
the implementation and the bound it realizes. We investigate this tradeoff in
the general case and prove upper bounds, which reduce the search space for the
algorithm, and matching lower bounds.
