Parameterized Complexity and Kernel Bounds for Hard Planning Problems

The propositional planning problem is a notoriously difficult computational problem. Downey et al. (1999) initiated the parameterized analysis of planning (with plan length as the parameter) and B"ackstr"om et al. (2012) picked up this line of research and provided an extensive parameterized analysis under various restrictions, leaving open only one stubborn case. We continue this work and provide a full classification. In particular, we show that the case when actions have no preconditions and at most $e$ postconditions is fixed-parameter tractable if $e\leq 2$ and W[1]-complete otherwise. We show fixed-parameter tractability by a reduction to a variant of the Steiner Tree problem; this problem has been shown fixed-parameter tractable by Guo et al. (2007). If a problem is fixed-parameter tractable, then it admits a polynomial-time self-reduction to instances whose input size is bounded by a function of the parameter, called the kernel. For some problems, this function is even polynomial which has desirable computational implications. Recent research in parameterized complexity has focused on classifying fixed-parameter tractable problems on whether they admit polynomial kernels or not. We revisit all the previously obtained restrictions of planning that are fixed-parameter tractable and show that none of them admits a polynomial kernel unless the polynomial hierarchy collapses to its third level.

💡 Research Summary

The paper delivers a complete parameterized complexity classification for a long‑standing open case in propositional planning. The authors focus on planning instances where every action has no preconditions and at most e postconditions (effects). The parameter under study is the plan length k, i.e., the minimum number of actions required to reach the goal from the initial state.

Main results

1. Fixed‑parameter tractability for e ≤ 2 – When each action can affect at most two variables, the authors show that the planning problem can be reduced in polynomial time to a restricted version of the Steiner Tree problem. In the constructed graph, goal variables become terminals, and each action corresponds to an edge (or a tiny subgraph) connecting at most two terminals. The Steiner Tree variant is known to be FPT with respect to the number of terminals, thanks to the algorithm of Guo, Niedermeier, and Raible (2007). Consequently, the planning problem is solvable in time O(2^{O(k)}·poly(n)), establishing that it belongs to the class FPT.

2. **W