Parameterized Complexity and Kernel Bounds for Hard Planning Problems

TL;DR

This paper classifies the parameterized complexity of certain planning problems, showing fixed-parameter tractability for some cases and proving the non-existence of polynomial kernels for all fixed-parameter tractable cases unless a major complexity collapse occurs.

Contribution

It provides a complete classification of the parameterized complexity of planning with no preconditions and limited postconditions, and establishes kernelization lower bounds for these problems.

Findings

Planning with actions having no preconditions and at most 2 postconditions is fixed-parameter tractable.

The problem becomes W[1]-complete when the number of postconditions exceeds 2.

None of the fixed-parameter tractable planning problems admits a polynomial kernel unless the polynomial hierarchy collapses.

Abstract

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.