Delete Relaxations for Planning with State-Dependent Action Costs

Supporting state-dependent action costs in planning admits a more compact representation of many tasks. We generalize the additive heuristic h and compute it by embedding decision-diagram representations of action cost functions into the RPG. We give a theoretical evaluation and present an implementation of the generalized h heuristic. This allows us to handle even the hardest instances of the combinatorial ACADEMIC ADVISING domain from the IPPC 2014. Introduction and Preliminaries State-dependent action costs (SDAC) often admit more compact domain representations than state-independent action costs. We show how action cost functions can be represented as edge-valued multi-valued decision diagrams (EVMDDs) (Ciardo and Siminiceanu 2002), which allows us to detect and exploit structure in the cost functions. EVMDDs can be used to derive a compact compilation to constant-cost actions or can directly be embedded into the relaxed planning graph (RPG). We define a natural generalization of the additive heuristic hadd to SDAC and show that the EVMDD embedding can be used to compute this generalized hadd heuristic. We apply this procedure to the ACADEMIC ADVISING domain (Guerin et al. 2012) and compare the results with the standard heuristic of the PROST planner (Keller and Eyerich 2012). We consider SAS tasks (Bäckström and Nebel 1995) with the usual syntax and semantics. Additionally, each action a has an associated cost function ca : D1 × · · · × Dn → N, where D1, . . . ,Dn are the finite domains of the variables on which ca depends. Let Sa be the set of valuations of those variables and let Fa be the set of facts (v, d) for v ∈ Sa and d in the domain of v. Without loss of generality, we assume that for each action a, the sets of variables mentioned in the precondition and those on which ca depends are disjoint. We may also view ca as a function over states. We use the usual definition of relaxed planning tasks and define the cost of an action a in a relaxed state s, ca(s) as the minimal cost ca(s) for any unrelaxed state s that is subsumed by s. Next, we give a generalization of the additive heuristic hadd (Bonet and Geffner 2001) A long version of this paper was accepted to IJCAI-15 (Geißer, Keller, and Mattmüller 2015). To cite this work, please cite the IJCAI-15 paper. to tasks with SDAC that correctly reflects action costs. Let s? be the goal description, let sp stand for a partial state and f = (v, d) for a fact, and let A(f) be the set of achievers of f . As in the classical setting, hadd(s) = hadd s (s?), and hadd s (sp) = ∑ f∈sp h add s (f). Unlike in the classical setting, hadd s (f) = { 0 if f ∈ s min a∈A(f) [ hadd s (pre(a)) + C a s ] else, where C s = min ŝ∈Sa [ca(ŝ) + h add s (ŝ)], where pre(a) is the precondition of a. We minimize over all achievers a of f and over all possible situations where a is applicable by replacing the constant action cost of the classical setting withC s . The central new challenge is the efficient computation of C s . The number of states in Sa is exponential in the number of variables on which ca depends, and C s cannot always be additively decomposed by these variables. However, we can represent ca and C s using EVMDDs such that ca and C s are efficiently computable in the size of the EVMDD representation and that the size of the representation itself is, although worst-case exponential, compact in many “typical”, well-behaved cases. Edge-Valued Decision Diagrams Each cost function ca : D1 × · · · × Dn → N over variables v1, . . . , vn with domains Dv = {0, . . . , |Dv| − 1} can be encoded as an EVMDD (Ciardo and Siminiceanu 2002). EVMDDs are directed acyclic graphs with unique source edges and sink nodes and with interior nodes branching over the domain values of the variables. Edges have associated weights, and for a concrete valuation s of the variables, the value E(s) of an EVMDD E is determined by traversing the unique path in E corresponding to s and summing up the edge weights along the way. To compute C s , we need to incorporate h add s values into Ea. To that end, on each path through Ea, each variable on which ca depends must be tested. Hence, in the following we assume that Ea includes branches over all variables on all paths, and call such an Ea quasi-reduced. Example 1. Consider the action cost function ca = AB + C + 2 with DA = DC = {0, 1}, and DB = {0, 1, 2}.