Planning is a notoriously hard combinatorial search problem. In many interesting domains, current planning algorithms fail to scale up gracefully. By combining a general, stochastic search algorithm and appropriate problem encodings based on propositional logic, we are able to solve hard planning problems many times faster than the best current planning systems. Although stochastic methods have been shown to be very e ective on a wide range of scheduling problems, this is the rst demonstration of its power on truly challenging classical planning instances. This work also provides a new perspective on representational issues in planning.

I present several computational complexity results for propositional STRIPS planning, i.e., STRIPS planning restricted to ground formulas. Different planning problems can be defined by restricting the type of formulas, placing limits on the number of pre- and postconditions, by restricting negation in pre- and postconditions, and by requiring optimal plans. For these types of restrictions, I show when planning is tractable (polynomial) and intractable (NPhard) . In general, it is PSPACE-complete to determine if a given planning instance has any solutions. Extremely severe restrictions on both the operators and the formulas are required to guarantee polynomial time or even NP-completeness. For example, when only ground literals are permitted, determining plan existence is PSPACE-complete even if operators are limited to two preconditions and two postconditions. When definite Horn ground formulas are permitted, determining plan existence is PSPACE-complete even if operators are limited t...

In this paper, we examine how the complexity of domain-independent planning with STRIPS-style operators depends on the nature of the planning operators. We show

In this paper, we show that in the best-known version of the blocks world (and several related versions), planning is difficult, in the sense that finding an optimal plan is NP-hard. However, the NP-hardness is not due to deleted-condition interactions, but instead due to a situation which we call a deadlock. For problems that do not contain deadlocks, there is a simple hill-climbing strategy that can easily find an optimal plan, regardless of whether or not the problem contains any deleted-condition interactions. The above result is rather surprising, since one of the primary roles of the blocks world in the planning literature has been to provide examples of deleted-condition interactions such as creative destruction and Sussman's anomaly. However, we can explain why deadlocks are hard to handle in terms of a domain-independent goal interaction which we call an enabling-condition interaction, in which an action invoked to achieve one goal has a side-effect of making it easier to achi...

The ability of a planner to reuse parts of old plans is hypothesized to be a valuable tool for improving efficiency of planning by avoiding the repetition of the same planning effort. We test this hypothesis from an analytical and empirical point of view. A comparative worst-case complexity analysis of generation and reuse under different assumptions reveals that it is not possible to achieve a provable efficiency gain of reuse over generation. Further, assuming "conservative" plan modification, plan reuse can actually be strictly more difficult than plan generation. While these results do not imply that there won't be an efficiency gain in some situations, retrieval of a good plan may present a serious bottleneck for plan reuse systems, as we will show. Finally, we present the results of an empirical study of two different plan reuse systems, pointing out possible pitfalls one should be aware of when attempting to employ reuse methods. ? This work was supported by the German Ministry...

This paper reports on experiments where techniques of supervised machine learning are applied to the problem of planning. The input to the learning algorithm is composed of a description of a planning domain, planning problems in this domain, and solutions for them. The output is an efficient algorithm --- a strategy --- for solving problems in that domain. We test the strategy on an independent set of planning problems from the same domain, so that success is measured by its ability to solve complete problems. A system, L2Act, has been developed in order to perform these experiments. We have experimented with the blocks world domain, and the logistics domain, using strategies in the form of a generalization of decision lists, where the rules on the list are existentially quantified first order expressions. The learning algorithm is a variant of Rivest`s [39] algorithm, improved with several techniques that reduce its time complexity. As the experiments demonstrate, generalization is a...

One of the more controversial recent planning algorithms is the SHOP algorithm, an HTN planning algorithm that plans for tasks in the same order that they are to be executed. SHOP can use domaindependent knowledge to generate plans very quickly, but it can be difficult to write good knowledge bases for SHOP. Our hypothesis is that this difficulty is because SHOP’s total-ordering requirement for the subtasks of its methods is more restrictive than it needs to be. To examine this hypothesis, we have developed a new HTN planning algorithm called SHOP2. Like SHOP, SHOP2 is sound and complete, and it constructs plans in the same order that they will later be executed. But unlike SHOP, SHOP2 allows the subtasks of each

Many planning problems have recently been shown to be inherently intractable. For example, finding the shortest plan in the blocksworld domain is NP-hard, and so is planning in even some of the most limited STRIPSstyle planning formalisms. We explore the question as to what extent these negative results can be attributed to the insistence on finding plans of minimal length. Using recent results form the theory of combinatorial optimization, we show that for domain-independent planning, one cannot efficiently generate any reasonable approximation of the optimal plan. Our result holds for a very restricted form of STRIPS. So, the negative complexity results for domainindependent planning are not just a consequence of searching for the optimal plans, because even finding reasonable approximations is hard. Next we consider domain-dependent planning. For blocks-world planning one can generate in polynomial time good approximations of the minimal plan --- within a factor of two of optimal. W...

This paper reports an analysis of near-optimal Blocks World planning. Various methods are clarified, and their time complexity is shown to be linear in the number of blocks, which improves their known complexity bounds. The speed of the implemented programs (ten thousand blocks are handled in a second) enables us to make empirical observations on large problems. These suggest that the above methods have very close average performance ratios, and yield a rough upper bound on those ratios well below the worst case of 2. Further, they lead to the conjecture that in the limit the simplest linear time algorithm could be just as good on average as the optimal one. Motivation The Blocks World (bw) is an artificial planning domain, of little practical interest. Nonetheless, we see at least two reasons for examining it in more detail. In the first place, for good or ill, bw is by far the most extensively used example in the planning literature. It often serves for demonstrating t...

The ability of a planner to modify a plan is considered as a valuable tool for improving efficiency of planning by avoiding the repetition of the same planning effort. From a computational complexity point of view, however, it is by no means obvious that modifying a plan is computationally as easy as planning from scratch if the modification has to follow the principle of "conservatism," i.e., to reuse as much of the old plan as possible. Indeed, considering propositional STRIPS planning, it turns out that conservative plan modification is as hard as planning and can sometimes be harder than plan generation. Furthermore, this holds even if we consider modification problems where the old and the new goal specification are similar. We put these results into perspective and discuss the relationship to existing plan modification systems. 1 Introduction Plan generation in complex domains is normally a resource and time consuming process. One way to improve the efficiency of planning syste...