Parameterized Complexity of Graph Constraint Logic

TL;DR

This paper investigates the parameterized complexity of graph constraint logic problems, demonstrating PSPACE-completeness under various graph parameters and extending the framework to reconfiguration problems and Rush Hour puzzles.

Contribution

It proves PSPACE-completeness of restricted NCL on graphs of bounded bandwidth and extends complexity results to reconfiguration problems and Rush Hour puzzles.

Findings

Restricted NCL remains PSPACE-complete on graphs of bounded bandwidth.

Reconfiguration versions of classical problems are PSPACE-complete on planar graphs of bounded bandwidth.

Rush Hour on $k\times n$ boards is PSPACE-complete even for small $k$.

Abstract

Graph constraint logic is a framework introduced by Hearn and Demaine, which provides several problems that are often a convenient starting point for reductions. We study the parameterized complexity of Constraint Graph Satisfiability and both bounded and unbounded versions of Nondeterministic Constraint Logic (NCL) with respect to solution length, treewidth and maximum degree of the underlying constraint graph as parameters. As a main result we show that restricted NCL remains PSPACE-complete on graphs of bounded bandwidth, strengthening Hearn and Demaine's framework. This allows us to improve upon existing results obtained by reduction from NCL. We show that reconfiguration versions of several classical graph problems (including independent set, feedback vertex set and dominating set) are PSPACE-complete on planar graphs of bounded bandwidth and that Rush Hour, generalized to $k\times n$ boards, is PSPACE-complete even when $k$ is at most a constant.