Solution Discovery via Reconfiguration for Problems in P

Authors Mario Grobler , Stephanie Maaz , Nicole Megow , Amer E. Mouawad , Vijayaragunathan Ramamoorthi , Daniel Schmand , Sebastian Siebertz



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Author Details

Mario Grobler
  • University of Bremen, Germany
Stephanie Maaz
  • University of Waterloo, Canada
Nicole Megow
  • University of Bremen, Germany
Amer E. Mouawad
  • American University of Beirut, Lebanon
Vijayaragunathan Ramamoorthi
  • University of Bremen, Germany
Daniel Schmand
  • University of Bremen, Germany
Sebastian Siebertz
  • University of Bremen, Germany

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Mario Grobler, Stephanie Maaz, Nicole Megow, Amer E. Mouawad, Vijayaragunathan Ramamoorthi, Daniel Schmand, and Sebastian Siebertz. Solution Discovery via Reconfiguration for Problems in P. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 76:1-76:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.76

Abstract

In the recently introduced framework of solution discovery via reconfiguration [Fellows et al., ECAI 2023], we are given an initial configuration of k 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 b 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 token 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) k and transformation budget (number of steps) b. 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Combinatorics
Keywords
  • solution discovery
  • reconfiguration
  • spanning tree
  • shortest path
  • matching
  • cut

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