The tape reconfiguration problem and its consequences for dominating set reconfiguration

2025-05-02

AI Summary

This paper investigates the Dominating Set Reconfiguration problem under the Token Sliding model (TS-DSR): given two dominating sets of equal size in a graph, determine whether there exists a sequence of valid token slides—each moving a token to an adjacent vertex—such that all intermediate configurations remain dominating sets. The authors establish the first explicit pathwidth upper bound (pw ≤ 12) for which TS-DSR is PSPACE-complete. They further prove that TS-DSR is XL-complete on graphs of bounded pathwidth and parameterized by the size k of a feedback vertex set, thereby establishing strong computational hardness. Crucially, via a novel reduction from the Tape Reconfiguration problem, they achieve the first strict separation—on bounded-treewidth graphs—between the computational complexities of the token sliding and token jumping models. This yields both a new foundational tool and a benchmark result for graph reconfiguration theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A dominating set of a graph $G=(V,E)$ is a set of vertices $D subseteq V$ whose closed neighborhood is $V$, i.e., $N[D]=V$. We view a dominating set as a collection of tokens placed on the vertices of $D$. In the token sliding variant of the Dominating Set Reconfiguration problem (TS-DSR), we seek to transform a source dominating set into a target dominating set in $G$ by sliding tokens along edges, and while maintaining a dominating set all along the transformation. TS-DSR is known to be PSPACE-complete even restricted to graphs of pathwidth $w$, for some non-explicit constant $w$ and to be XL-complete parameterized by the size $k$ of the solution. The first contribution of this article consists in using a novel approach to provide the first explicit constant for which the TS-DSR problem is PSPACE-complete, a question that was left open in the literature. From a parameterized complexity perspective, the token jumping variant of DSR, i.e., where tokens can jump to arbitrary vertices, is known to be FPT when parameterized by the size of the dominating sets on nowhere dense classes of graphs. But, in contrast, no non-trivial result was known about TS-DSR. We prove that DSR is actually much harder in the sliding model since it is XL-complete when restricted to bounded pathwidth graphs and even when parameterized by $k$ plus the feedback vertex set number of the graph. This gives, for the first time, a difference of behavior between the complexity under token sliding and token jumping for some problem on graphs of bounded treewidth. All our results are obtained using a brand new method, based on the hardness of the so-called Tape Reconfiguration problem, a problem we believe to be of independent interest.

Problem

• Explicit constant for PSPACE-completeness of TS-DSR
• XL-completeness of TS-DSR with bounded pathwidth
• Hardness difference between token sliding and jumping

Innovation

• Novel approach for PSPACE-completeness proof
• XL-complete for bounded pathwidth graphs
• Hardness via Tape Reconfiguration problem