New Complexity Results for Structurally Restricted Numeric Planning

Alexander Shleyfman, Daniel Gnad, Peter Jonsson

Keywords: numeric planning, complexity analysis, problem structure, causal dependencies

TL;DR: We analyze the complexity of simple numeric planning and show that it is NP-hard with a single (numeric) variable, decidable when all numeric variables are causal-graph leaf nodes, and lies in PSPACE if on top the propositional state space is fixed.

Abstract: Numeric planning is known to be undecidable even under severe restrictions. Prior work investigated the decidability boundaries by restricting the expressiveness of the planning formalism in terms of the numeric functions allowed in conditions and effects. In this work, we fix one specific such formalism, simple numeric planning (SNP), which, while only allowing linear conditions and action effects to add constants, is still undecidable. We analyze the complexity of SNP by (1) restricting the number of numeric variables, and (2) restricting the causal structure. First, we concentrate on numeric planning with exactly one (numeric) variable. We present a pseudo-polynomial algorithm to solve such tasks, and show NP-hardness and PSPACE-membership for the corresponding decision problem. Second, we restrict the interaction between variables in terms of the causal graph. As our main result, we show that SNP with an arbitrary number of numeric causal-graph leaf variables is decidable, and lies in PSPACE if the propositional state space has fixed size.