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Parameterized Complexity of Optimal Planning: a Detailed Map Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence Parameterized Complexity of Optimal Planning: A Detailed Map Martin Kronegger, Andreas Pfandler, and Reinhard Pichler Vienna University of Technology, Austria fkronegger, pfandler, [email protected] Abstract problem instance but also in terms of the size k of some pa- rameter (or combination of parameters) that describes certain The goal of this paper is a systematic parameter- characteristics of the given problem instance. The primary ized complexity analysis of different variants of goal of a parameterized complexity analysis is to identify propositional STRIPS planning. We identify sev- fixed-parameter tractability (FPT), i.e., the problem can be eral natural problem parameters and study all pos- solved in time f(k) · nO(1), where n denotes the size of the sible combinations of 9 parameters in 6 differ- problem instance and f(k) depends on the parameter k only ent settings. These settings arise, for instance, (but not on n). The exponential time complexity is thus con- from the distinction if negative effects of actions fined to the parameter, i.e., the function f(k). Moreover, an are allowed or not. We provide a complete pic- FPT result immediately yields tractability of the problem if ture by establishing for each case either paraNP- the parameter is bounded by a constant. In the area of plan- hardness (i.e., the parameter combination does not ning, very few FPT results are known. Downey et al. [1999] help) or W[t]-completeness with t 2 f1; 2g (i.e., proved FPT for STRIPS planning when parameterized by the fixed-parameter intractability), or FPT (i.e., fixed- plan length and the treewidth of an instance. Backstr¨ om¨ et al. parameter tractability). [2012] proved FPT of SAS+ planning with restriction P re- called above when considering the plan length as parameter. 1 Introduction In principle, any (natural) characteristic of problem in- Planning is a classical reasoning task in AI. In general, it stances is worth considering as parameter. Of course, there is computationally hard. Bylander [1994] provided a com- is no guarantee that a particular parameter yields a tractable prehensive complexity analysis of propositional STRIPS (be- fragment. If restricting a parameter by a constant does not low simply referred to as STRIPS, for short) – a fundamental lead to tractability, we call the problem paraNP-hard. By- model of planning. Without any restrictions (in particular, im- lander [1994] showed that STRIPS remains NP-hard even if posing no restriction on the plan length), STRIPS is PSPACE- every action has only one precondition and one effect. In complete. With a polynomial upper bound on the plan length, other words, the problem is paraNP-hard w.r.t. the parameter the problem becomes NP-complete. Bylander also identified combination “precondition size and effect size”. several restrictions which make STRIPS planning tractable, In parameterized complexity, the area between the “nice” e.g., by allowing only positive preconditions and only one case of FPT and the extreme case of paraNP-hardness is fluent (i.e., variable) to be affected by each action. even more fine-grained: a whole hierarchy of complexity Many attempts have been made to identify further tractable classes W[1], W[2], etc. lies in between and is used to classify fragments of planning. Backstr¨ om¨ and Klein [1991] have in- fixed-parameter intractable problems. For instance, STRIPS troduced various restrictions for the more general planning planning parameterized by the plan length was shown W[1]- formalism SAS+, where the domain is not necessarily binary. hard by Downey et al. [1999] and actually W[2]-complete These restrictions include, e.g., P (i.e., every fluent is set to by Backstr¨ om¨ et al. [2012]. It is a commonly believed a particular value by at most one action) and U (i.e., only complexity-theoretic assumption that FPT 6= W[1], which one fluent to be affected by an action). Backstr¨ om¨ and Nebel is also supported by the exponential time hypothesis (ETH). [1995] investigated the complexity under all combinations of Hence showing hardness for W[1] (or higher classes) rules these restrictions. Gimenez´ and Jonsson [2008], Chen and out the existence of an FPT-algorithm. Indeed, for W[t]- Gimenez´ [2010], and Katz and Domshlak [2008] analyzed complete problems, only algorithms with run time O(nf(k)) planning under various restrictions of the causal graph. are known, i.e., the parameter k occurs in the exponent of the A modern approach for identifying tractable fragments input size n. This is worse than the upper bound f(k) · nO(1) of hard problems comes from parameterized complexity for FPT, but it still allows us to identify a PTIME-solvable [Downey and Fellows, 1999], which introduces a multivari- fragment of the problem by imposing a constant upper bound ate view of complexity. The time needed to solve a problem on the parameter value. is thus measured not only in terms of the mere size n of a In various areas of reasoning, systematic parameterized 954 complexity analyses (by considering all combinations of sev- where only positive literals are allowed in the effect of the eral problem parameters) have been conducted recently. For actions. The special case of PSN where only positive literals instance, the parameterized complexity of abduction was are allowed in the precondition of the actions and in the goal studied by Fellows et al. [2012], of circumscription by Lack- is usually referred to as (classical propositional) STRIPS. ner and Pfandler [2012a], and of handling minimal models An action a 2 A is applicable in state s if pre(a) is satisfied by Lackner and Pfandler [2012b]. For planning, very little by s, when s is considered as an interpretation. A conjunc- is known about the parameterized complexity apart from the tion of literals has a unique model when considering only the few results recalled above. The goal of this paper is to close variables which actually occur in this conjunction. We shall this gap by embarking on a systematic parameterized com- refer to this unique model when we speak of the model of the plexity analysis of planning. goal, the model of an effect, etc. Let m be the model of eff(a) We start our analysis by identifying several natural param- for some action a. For the resulting state s0 after executing an eters of planning instances. All of these parameters can be applicable action a 2 A in state s we have s0(v) = m(v) if simply read off from an instance. Some of them describe the v 2 var (eff(a)) and s0(v) = s(v) otherwise. instance, e.g., size of the preconditions, size of the effects, Let π = [a1; :::; al] be a sequence of actions and s0; : : : ; sl maximum number of occurrences of each fluent, etc. The be states. Then π is called a plan from s0 to sl if ai 2 π other parameters refer to desired properties of the solutions: is applicable in state si−1 and si is the resulting state after 0 the plan length as has already been studied before and a re- executing ai in state si−1 for i 2 [l]. A state s is called finement of it, namely the total number of fluent changes in reachable from a state s if there is a plan from s to s0. Let m a plan. We study all possible combinations of 9 parameters be the model of G and s a state. Then s is called a goal state in 6 different settings. These settings arise depending if neg- if s(v) = m(v) for all v 2 var (G). We call π a plan for the ative preconditions/effects of actions are allowed or not and planning specification (V; A; I; G) if s0 = I, and sl is a goal if we consider the length or the number of fluent changes as state. The plan length of π is its number of actions l. a possible restriction on allowed plans. In principle we thus get 6 · 29 = 3072 cases. However, in Section 3 we shall iden- Parameterized Complexity. Parameterized algorithmics tify various dependencies between these parameters so that (cf. [Downey and Fellows, 1999; Flum and Grohe, 2006; we only need to establish the parameterized complexity of a Niedermeier, 2006]) is a promising approach to obtain effi- small subset of these cases to cover all of them. For details, cient algorithms for fragments of NP-hard problems. In a pa- see Section 3. rameterized complexity analysis the runtime of an algorithm is studied with respect to a parameter k 2 and input size n. Contribution. We provide a complete picture of the pa- N The basic idea here is to find a parameter that describes the rameterized complexity of three variants of propositional structure of the instance such that the combinatorial explosion STRIPS planning. For each of the 3072 cases we estab- can be confined to this parameter. The most favourable class lish either paraNP-hardness (i.e., the parameter combination is FPT (fixed-parameter tractable) which contains problems does not help) or W[t]-completeness with t 2 f1; 2g (i.e., that can be decided by an algorithm running in f(k) · nO(1) fixed-parameter intractability), or FPT (i.e., fixed-parameter time, where f is a computable function. We call such an al- tractability). FPT is achieved, e.g., for the combined pa- gorithm fixed-parameter tractable (fpt). rameters plan length and maximum number of occurrences ∗ of each fluent. This underlines the importance of studying Formally, a parameterized problem is a subset of Σ × N, combinations of parameters, since the first parameter alone where Σ is the input alphabet.
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