Structure and Problem Hardness: Goal Asymmetry and DPLL Proofs in SAT-Based Planning

Abstract

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.

Friends or Foes? On Planning as Satisfiability and Abstract CNF Encodings

“…For completeness, we include a proof in Appendix A, based on "folklore" ideas from proof complexity literature. An alternative proof, with a somewhat different notation, may also be found in the appendix of a recent article by Hoffmann, Gomes, and Selman (2007).…”

“…For example, in Gripper, the available time steps serve as "holes" and the actions picking/dropping balls are the "pigeons" (for a related investigation, see Hoffmann et al, 2007). It also seems quite natural that a planning task may consist of two disconnected parts, one of which is complex while the other one is easy to prove unsolvable in the given number of steps.…”

Tradeoffs in the Complexity of Backdoor Detection

“…The MAP problem domain is a synthetic logistics planning domain for which the size of the strong UP-backdoors is well understood [13]. In this domain, n is the number of nodes in the map graph and k is the number of locations to visit.…”

“…All MAP instances considered are unsatisfiable, encoding one planning step less than the length of the optimal plan. Hoffmann et al [13] identify that MAP instances with k = 2n − 3 (called asymmetric) have logarithmic size DPLL refutations (and backdoors). We evaluate the size of the backdoors in asymmetric MAP instances of various sizes (n = 5..50).…”

“…For the car configuration problem, strong SATZ-backdoor sets involve 0-0.7% of the variables. We next consider synthetic logistics planning instances over n variables that are known to have strong UP-backdoors of size log n [13]. For all these instances, the size of the strong SATZ-backdoor sets is either zero or one.…”