Unavoidable dead-ends are common in many probabilistic planning problems, e.g. when actions may fail or when operating under resource constraints. An important objective in such settings is MaxProb, determining the maximal probability with which the goal can be reached, and a policy achieving that probability. Yet algorithms for MaxProb probabilistic planning are severely underexplored, to the extent that there is scant evidence of what the empirical state of the art actually is. We close this gap with a comprehensive empirical analysis. We design and explore a large space of heuristic search algorithms, systematizing known algorithms and contributing several new algorithm variants. We consider MaxProb, as well as weaker objectives that we baptize AtLeastProb (requiring to achieve a given goal probabilty threshold) and ApproxProb (requiring to compute the maximum goal probability up to a given accuracy). We explore both the general case where there may be 0-reward cycles, and the practically relevant special case of acyclic planning, such as planning with a limited action-cost budget. We design suitable termination criteria, search algorithm variants, dead-end pruning methods using classical planning heuristics, and node selection strategies. We design a benchmark suite comprising more than 1000 instances adapted from the IPPC, resource-constrained planning, and simulated penetration testing. Our evaluation clarifies the state of the art, characterizes the behavior of a wide range of heuristic search algorithms, and demonstrates significant benefits of our new algorithm variants.

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Abstract. In Verification and in (optimal) AI Planning, a successful method is to formulate the application as boolean satisfiability (SAT), and solve it with state-of-the-art DPLL-based procedures. There is a lack of understanding of why this works so well. Focussing on the Planning context, we identify a form of problem structure concerned with the symmetrical or asymmetrical nature of the cost of achieving the individual planning goals. We quantify this sort of structure with a simple numeric parameter called AsymRatio, ranging between 0 and 1. We run experiments in 10 benchmark domains from the International Planning Competitions since 2000; we show that AsymRatio is a good indicator of SAT solver performance in 8 of these domains. We then examine carefully crafted synthetic planning domains that allow control of the amount of structure, and that are clean enough for a rigorous analysis of the combinatorial search space. The domains are parameterized by size, and by the amount of structure. The CNFs we examine are unsatisfiable, encoding one planning step less than the length of the optimal plan. We prove upper and lower bounds on the size of the best possible DPLL refutations, under different settings of the amount of structure, as a function of size. We also identify the best possible sets of branching variables (backdoors). With minimum AsymRatio, we prove exponential lower bounds, and identify minimal backdoors of size linear in the number of variables.With maximum AsymRatio, we identify logarithmic DPLL refutations (and backdoors), showing a doubly exponential gap between the two structural extreme cases. The reasons for this behavior -the proof arguments -illuminate the prototypical patterns of structure causing the empirical behavior observed in the competition benchmarks.

A successful partial delete relaxation method is to compute hFF in a compiled planning task PiC which represents a set C of 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. We provide a direct characterization of h+(PiC), without compilation, making explicit how it arises from a "marriage" of the critical-path heuristic hm with (a somewhat novel view of) h+. This explicit view allows us to derive a direct characterization of hFF(PiC), which in turn allows us to compute a version of that heuristic function in time polynomial in |C|.

Merge-and-shrink abstraction (M&S) is an approach for constructing admissible heuristic functions for cost-optimal planning. It enables the targeted design of abstractions, by allowing to choose individual pairs of (abstract) states to aggregate into one. A key question is how to actually make these choices, so as to obtain an informed heuristic at reasonable computational cost. Recent work has addressed this via the well-known notion of bisimulation. When aggregating only bisimilar states -- essentially, states whose behavior is identical under every planning operator -- M&S yields a perfect heuristic. However, bisimulations are typically exponentially large. Thus we must relax the bisimulation criterion, so that it applies to more state pairs, and yields smaller abstractions. We herein devise a fine-grained method for doing so. We restrict the bisimulation criterion to consider only a subset K of the planning operators. We show that, if K is chosen appropriately, then M&S still yields a perfect heuristic, while abstraction size may decrease exponentially. Designing practical approximations for K, we obtain M&S heuristics that are competitive with the state of the art.