  We formalize the problem of selecting the optimal set of options for planning
as that of computing the smallest set of options so that planning converges in
less than a given maximum of value-iteration passes. We first show that the
problem is NP-hard, even if the task is constrained to be deterministic---the
first such complexity result for option discovery. We then present the first
polynomial-time boundedly suboptimal approximation algorithm for this setting,
and empirically evaluate it against both the optimal options and a
representative collection of heuristic approaches in simple grid-based domains
including the classic four-rooms problem.
