A Structural Complexity Analysis of Hierarchical Task Network Planning

TL;DR

This paper studies the structural complexity of Hierarchical Task Network (HTN) planning under natural restrictions on task networks. It defines primitive task networks and measures such as generalized partial order width ($\text{gpow}$) and compound-task parameters ($C_d$, $C_s$, $C_\#$, $C_c$), and analyzes three core problems: Plan Verification, Plan Existence, and State Reachability. It proves polynomial-time solvability for primitive networks with bounded $\text{gpow}$ and develops an algorithmic meta-theorem showing how this tractability lifts to compound HTNs when $C_d$, $C_s$, $C_\#$ (and a stability measure) are bounded, with tight lower bounds. The paper also provides a nuanced parameterized complexity analysis, establishing W-hardness for width-based parameterizations while giving fixed-parameter tractable results parameterized by vertex cover number and extended metatheorems for compounds. Together, these results map a rich landscape of tractability and hardness, offering constructive DP/ILP approaches and guiding future HTN complexity research.

Abstract

We perform a refined complexity-theoretic analysis of three classical problems in the context of Hierarchical Task Network Planning: the verification of a provided plan, whether an executable plan exists, and whether a given state can be reached. Our focus lies on identifying structural properties which yield tractability. We obtain new polynomial algorithms for all three problems on a natural class of primitive networks, along with corresponding lower bounds. We also obtain an algorithmic meta-theorem for lifting polynomial-time solvability from primitive to general task networks, and prove that its preconditions are tight. Finally, we analyze the parameterized complexity of the three problems.

Figures (3)

• Figure 1: Overview of our results and respective theorems. Generalized Partial Order Width is abbreviated to gpow, and Vertex Cover Number to vcn. Results in gray boxes refer to primitive instances.
• Figure 2: The digraph $D_\prec$ for a primitive HTN with tasks $T={\{t_1,t_2,t_3,t_4\}}$, actions $A={\{a_1,a_2,a_3\}}$, propositions $F=\{1,2\}$, and initial state $s_0 = \emptyset$. A task's action is in its node. There is an arc from a task $t$ to a task $t'$ if $(t,t')\in\prec$. Here, the only sequences that execute all tasks are $t_1,t_2,t_3,t_4$ and $t_1,t_3,t_2,t_4$.
• Figure 3: Constructed task network for reductions from Shuffle Product with words $c_1,\ldots, c_w$ and $u$. Each word is represented by a chain of tasks, where each task represents a letter in that word. The two letters in the alphabet are represented in orange and blue.

Theorems & Definitions (36)

• Theorem 1
• proof
• Theorem 2
• proof : Proof Sketch
• Theorem 3
• proof
• Theorem 4
• proof : Proof Sketch
• Theorem 5
• proof : Proof Sketch