Solution discovery via reconfiguration for problems in P
Abstract
In the recently introduced framework of solution discovery via reconfiguration [Fellows et al., ECAI 2023], we are given an initial configuration of tokens on a graph and the question is whether we can transform this configuration into a feasible solution (for some problem) via a bounded number of small modification steps. In this work, we study solution discovery variants of polynomial-time solvable problems, namely Spanning Tree Discovery, Shortest Path Discovery, Matching Discovery, and Vertex/Edge Cut Discovery in the unrestricted token addition/removal model, the token jumping model, and the token sliding model. In the unrestricted token addition/removal model, we show that all four discovery variants remain in P. For the toking jumping model we also prove containment in P, except for Vertex/Edge Cut Discovery, for which we prove NP-completeness. Finally, in the token sliding model, almost all considered problems become NP-complete, the exception being Spanning Tree Discovery, which remains polynomial-time solvable. We then study the parameterized complexity of the NP-complete problems and provide a full classification of tractability with respect to the parameters solution size (number of tokens) and transformation budget (number of steps) . Along the way, we observe strong connections between the solution discovery variants of our base problems and their (weighted) rainbow variants as well as their red-blue variants with cardinality constraints.
Cite
@article{arxiv.2311.13478,
title = {Solution discovery via reconfiguration for problems in P},
author = {Mario Grobler and Stephanie Maaz and Nicole Megow and Amer E. Mouawad and Vijayaragunathan Ramamoorthi and Daniel Schmand and Sebastian Siebertz},
journal= {arXiv preprint arXiv:2311.13478},
year = {2023}
}