Combining the Delete Relaxation with Critical-Path Heuristics: A Direct Characterization

Abstract

Recent work has shown how to improve delete relaxation heuristics by computing relaxed plans, i.e., the hFF heuristic, in a compiled planning task PiC which represents a given set C of fact conjunctions explicitly. While this compilation view of such partial delete relaxation is simple and elegant, its meaning with respect to the original planning task is opaque, and the size of PiC grows exponentially in |C|. We herein provide a direct characterization, without compilation, making explicit how the approach arises from a combination of the delete-relaxation with critical-path heuristics. Designing equations characterizing a novel view on h+ on the one hand, and a generalized version hC of hm on the other hand, we show that h+(PiC) can be characterized in terms of a combined hcplus equation. This naturally generalizes the standard delete-relaxation framework: understanding that framework as a relaxation over singleton facts as atomic subgoals, one can refine the relaxation by using the conjunctions C as atomic subgoals instead. Thanks to this explicit view, we identify the precise source of complexity in hFF(PiC), namely maximization of sets of supported atomic subgoals during relaxed plan extraction, which is easy for singleton-fact subgoals but is NP-complete in the general case. Approximating that problem greedily, we obtain a polynomial-time hCFF version of hFF(PiC), superseding the PiC compilation, and superseding the modified PiCce compilation which achieves the same complexity reduction but at an information loss. Experiments on IPC benchmarks show that these theoretical advantages can translate into empirical ones.

Online Relaxation Refinement for Satisficing Planning: On Partial Delete Relaxation, Complete Hill-Climbing, and Novelty Pruning

“…The concept of partial delete relaxation aims to improve the accuracy of delete relaxation heuristics by taking some delete information into account. One such technique is based on explicit conjunctions, where a given set of conjunctions (fact sets) C are treated as atomic, and the facts contained in a conjunction c ∈ C must be achieved simultaneously (Haslum, 2012;Keyder, Hoffmann, & Haslum, 2012;Keyder et al, 2014;Hoffmann & Fickert, 2015;Fickert et al, 2016). The h CFF heuristic and its idealized counterpart h C+ compute such C-relaxed plans: Whenever a conjunction c ∈ C is a subset of the preconditions of an action, the partially relaxed plan π[h CFF ] must satisfy c instead of the individual facts contained in c. A conjunction c can only be achieved by an action a if a achieves some part of the conjunction (add (a)∩c = ∅) and does not delete another (del (a)∩c = ∅), and the remaining facts of c that are not added by a (c \ add (a)) are treated as additional preconditions.…”

“…Most planning heuristics can compute their estimates at different levels of precision: Abstraction heuristics (e.g., Clarke, Grumberg, & Long, 1994;Culberson & Schaeffer, 1998;Edelkamp, 2001;Helmert, Haslum, & Hoffmann, 2007;Helmert, Haslum, Hoffmann, & Nissim, 2014;Seipp & Helmert, 2018) construct an abstract state space, which can range from just a single state (where all heuristic estimates would be zero) to the full state space of the input task (computing the perfect heuristic h * ). Critical-path heuristics (Haslum & Geffner, 2000;Haslum, 2006;Fickert, Hoffmann, & Steinmetz, 2016) compute their estimates based on the most costly subgoals toward the goal, where considering larger subgoals results in a more accurate heuristic. Partial delete relaxation heuristics (Keyder, Hoffmann, & Haslum, 2014;Domshlak, Hoffmann, & Katz, 2015;Fickert et al, 2016) ignore some of the delete effects of the input task, interpolating between the full delete relaxation and non-relaxed semantics.…”

“…Critical-path heuristics (Haslum & Geffner, 2000;Haslum, 2006;Fickert, Hoffmann, & Steinmetz, 2016) compute their estimates based on the most costly subgoals toward the goal, where considering larger subgoals results in a more accurate heuristic. Partial delete relaxation heuristics (Keyder, Hoffmann, & Haslum, 2014;Domshlak, Hoffmann, & Katz, 2015;Fickert et al, 2016) ignore some of the delete effects of the input task, interpolating between the full delete relaxation and non-relaxed semantics.…”

“…The main advantage of the CEGAR approach is that it focuses the refinement on areas where the heuristic is currently flawed, making the refinement procedure more effective. In planning, CEGAR is used as the refinement method for Cartesian abstractions (Seipp & Helmert, 2013, 2018, pattern database heuristics (Rovner, Sievers, & Helmert, 2019), and the partial delete relaxation heuristic h CFF based on atomic conjunctions (Keyder et al, 2014;Fickert et al, 2016).…”

Merge-and-Shrink: A Compositional Theory of Transformations of Factored Transition Systems

“…In the merge-andshrink literature, such overlapping merging has been discussed hypothetically under the name of non-orthogonal merging (Helmert et al, 2007(Helmert et al, , 2014 but is not covered by the existing theory or implementations. A related technique somewhat more distant from the merge-and-shrink literature is the tracking of conjunctions of state variables in the h m and h C family of heuristics (e.g., Haslum & Geffner, 2000;Haslum, 2009Haslum, , 2012Keyder, Hoffmann, & Haslum, 2012;Fickert, Hoffmann, & Steinmetz, 2016).…”