We present three new complexity results for classes of planning problems with simple causal graphs. First, we describe a polynomial-time algorithm that uses macros to generate plans for the class 3S of planning problems with binary state variables and acyclic causal graphs. This implies that plan generation may be tractable even when a planning problem has an exponentially long minimal solution. We also prove that the problem of plan existence for planning problems with multi-valued variables and chain causal graphs is NP-hard. Finally, we show that plan existence for planning problems with binary state variables and polytree causal graphs is NP-complete.
Deep Dive into The Complexity of Planning Problems With Simple Causal Graphs.
We present three new complexity results for classes of planning problems with simple causal graphs. First, we describe a polynomial-time algorithm that uses macros to generate plans for the class 3S of planning problems with binary state variables and acyclic causal graphs. This implies that plan generation may be tractable even when a planning problem has an exponentially long minimal solution. We also prove that the problem of plan existence for planning problems with multi-valued variables and chain causal graphs is NP-hard. Finally, we show that plan existence for planning problems with binary state variables and polytree causal graphs is NP-complete.
Planning is an area of research in artificial intelligence that aims to achieve autonomous control of complex systems. Formally, the planning problem is to obtain a sequence of transformations for moving a system from an initial state to a goal state, given a description of possible transformations. Planning algorithms have been successfully used in a variety of applications, including robotics, process planning, information gathering, autonomous agents and spacecraft mission control. Research in planning has seen significant progress during the last ten years, in part due to the establishment of the International Planning Competition.
An important aspect of research in planning is to classify the complexity of solving planning problems. Being able to classify a planning problem according to complexity makes it possible to select the right tool for solving it. Researchers usually distinguish between two problems: plan generation, the problem of generating a sequence of transformations for achieving the goal, and plan existence, the problem of determining whether such a sequence exists. If the original STRIPS formalism is used, plan existence is undecidable in the first-order case (Chapman, 1987) and PSPACE-complete in the propositional case (Bylander, 1994). Using PDDL, the representation language used at the International Planning Competition, plan existence is EXPSPACE-complete (Erol, Nau, & Subrahmanian, 1995). However, planning problems usually exhibit structure that makes them much easier to solve. Helmert (2003) showed that many of the benchmark problems used at the International Planning Competition are in fact in P or NP.
A common type of structure that researchers have used to characterize planning problems is the so called causal graph (Knoblock, 1994). The causal graph of a planning problem is a graph that captures the degree of independence among the state variables of the problem, and is easily constructed given a description of the problem transformations. The independence between state variables can be exploited to devise algorithms for efficiently solving the planning problem. The causal graph has been used as a tool for describing tractable subclasses of planning problems (Brafman & Domshlak, 2003;Jonsson & Bäckström, 1998;Williams & Nayak, 1997), for decomposing planning problems into smaller problems (Brafman & Domshlak, 2006;Jonsson, 2007;Knoblock, 1994), and as the basis for domain-independent heuristics that guide the search for a valid plan (Helmert, 2006).
In the present work we explore the computational complexity of solving planning problems with simple causal graphs. We present new results for three classes of planning problems studied in the literature: the class 3S (Jonsson & Bäckström, 1998), the class C n (Domshlak & Dinitz, 2001), and the class of planning problems with polytree causal graphs (Brafman & Domshlak, 2003). In brief, we show that plan generation for instances of the first class can be solved in polynomial time using macros, but that plan existence is not solvable in polynomial time for the remaining two classes, unless P = NP. This work first appeared in a conference paper (Giménez & Jonsson, 2007); the current paper provides more detail and additional insights as well as new sections on plan length and CP-nets.
A planning problem belongs to the class 3S if its causal graph is acyclic and all state variables are either static, symmetrically reversible or splitting (see Section 3 for a precise definition of these terms). The class 3S was introduced and studied by Jonsson and Bäckström (1998) as an example of a class for which plan existence is easy (there exists a polynomial-time algorithm that determines whether or not a particular planning problem of that class is solvable) but plan generation is hard (there exists no polynomial-time algorithm that generates a valid plan for every planning problem of the class). More precisely, Jonsson and Bäckström showed that there are planning problems of the class 3S for which every valid plan is exponentially long. This clearly prevents the existence of an efficient plan generation algorithm.
Our first contribution is to show that plan generation for 3S is in fact easy if we are allowed to express a valid plan using macros. A macro is simply a sequence of operators and other macros. We present a polynomial-time algorithm that produces valid plans of this form for planning problems of the class 3S. Namely, our algorithm outputs in polynomial time a system of macros that, when executed, produce the actual valid plan for the planning problem instance. The algorithm is sound and complete, that is, it generates a valid plan if and only if one exists. We contrast our algorithm to the incremental algorithm proposed by Jonsson and Bäckström (1998), which is polynomial in the size of the output.
We also investigate the complexity of the class C n of planning problems with multivalued state variables and chain causal graphs. In
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