On the Computational Complexity of Stackelberg Planning and Meta-Operator Verification: Technical Report

TL;DR

This work provides the first formal complexity analysis of Stackelberg planning, showing that the general problem is $PSPACE$-complete and becomes $\Sigma^{\textrm{P}}_{2}$-complete when restricting plan length to be polynomial, implying potential exponential blow-up for compilations to classical planning. It analyzes tractable fragments under Bylander-style syntactic restrictions and reveals that Stackelberg planning can remain intractable even when classical planning is tractable. The paper further studies meta-operator verification, proving $PSPACE$-complete in general and $\Pi^{\textrm{P}}_{2}$-complete under polynomial plan-length bounds, highlighting fundamental limits on reductions and heuristic approaches. Collectively, these results clarify the complexity landscape of Stackelberg planning and its related verification tasks, informing the feasibility of compile-time reductions and the design of leader-follower heuristics.

Abstract

Stackelberg planning is a recently introduced single-turn two-player adversarial planning model, where two players are acting in a joint classical planning task, the objective of the first player being hampering the second player from achieving its goal. This places the Stackelberg planning problem somewhere between classical planning and general combinatorial two-player games. But, where exactly? All investigations of Stackelberg planning so far focused on practical aspects. We close this gap by conducting the first theoretical complexity analysis of Stackelberg planning. We show that in general Stackelberg planning is actually no harder than classical planning. Under a polynomial plan-length restriction, however, Stackelberg planning is a level higher up in the polynomial complexity hierarchy, suggesting that compilations into classical planning come with a worst-case exponential plan-length increase. In attempts to identify tractable fragments, we further study its complexity under various planning task restrictions, showing that Stackelberg planning remains intractable where classical planning is not. We finally inspect the complexity of meta-operator verification, a problem that has been recently connected to Stackelberg planning.

Theorems & Definitions (18)

• Definition 1: STACKELSAT
• Definition 2: STACKELMIN
• Proposition 1
• Theorem 1
• Theorem 2
• Definition 3: Polynomial Stackelberg Decision
• Theorem 3
• Definition 4
• Theorem 4
• Theorem 5