We have previously reported a number of tractable planning problems defined in the SAS+ formalism. This article complements these results by providing a complete map over the complexity of SAS+ planning under all combinations of the previously considered restrictions. We analyze the complexity of both finding a minimal plan and finding any plan. In contrast to other complexity surveys of planning, we study not only the complexity of the decision problems but also the complexity of the generation problems. We prove that the SAS+-PUS problem is the maximal tractable problem under the restrictions we have considered if we want to generate minimal plans. If we are satisfied with any plan, then we can generalize further to the SAS+-US problem, which we prove to be the maximal tractable problem in this case.

This article studies the problem of modifying the action ordering of a plan in order to optimise the plan according to various criteria. One of these criteria is to make a plan less constrained and the other is to minimize its parallel execution time. Three candidate definitions are proposed for the first of these criteria, constituting a sequence of increasing optimality guarantees. Two of these are based on deordering plans, which means that ordering relations may only be removed, not added, while the third one uses reordering, where arbitrary modifications to the ordering are allowed. It is shown that only the weakest one of the three criteria is tractable to achieve, the other two being NP-hard and even difficult to approximate. Similarly, optimising the parallel execution time of a plan is studied both for deordering and reordering of plans. In the general case, both of these computations are NP-hard. However, it is shown that optimal deorderings can be computed in polynomial time for a class of planning languages based on the notions of producers, consumers and threats, which includes most of the commonly used planning languages. Computing optimal reorderings can potentially lead to even faster parallel executions, but this problem remains NP-hard and difficult to approximate even under quite severe restrictions

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This article describes a polynomial-time, O( n'), planning algorithm for a limited class of planning problems.Compared to previous work on complexity of algorithms for knowledge-based or logic-based planning, our algorithm achieves computational tractability, but at the expense of only applying to a significantly more limited class of problems. Our algorithm is proven correct, and it always returns a parallel minimal plan if there is a plan at all.Cet article dtcrit un algorithme de planification de temps polynomial O ( n 3 ) pour une classe restreinte de problkmes de planification. Contrairement aux travaux precedents sur la complexitt des algorithmes pour la planification baste sur la logique ou les connaissances, I'algorithme dont il est question dans cet article permet d'obtenir la tractabilitk computationnelle; cependant, il ne peut Etre appliquC qu'a une categorie beaucoup plus restreinte de problkmes. Cet algorithme s'est donc rtvtlt correct et il genere toujours un plan minimal en parallkle lorsqu'il y en a un.