Solution Discovery via Reconfiguration for Problems in P

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

Thumbnail PDF


  • Filesize: 0.79 MB
  • 20 pages

Document Identifiers

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

Cite AsGet BibTex

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)


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
  • solution discovery
  • reconfiguration
  • spanning tree
  • shortest path
  • matching
  • cut


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM, 42(4):844-856, 1995. Google Scholar
  2. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12(2):308-340, 1991. Google Scholar
  3. Xuqing Bai, Renying Chang, and Xueliang Li. More on rainbow disconnection in graphs. Discussiones Mathematicae Graph Theory, 42:1185-1204, 2020. Google Scholar
  4. Paul Bonsma. The complexity of rerouting shortest paths. Theoretical computer science, 510:1-12, 2013. Google Scholar
  5. Nicolas Bousquet, Amer E. Mouawad, Naomi Nishimura, and Sebastian Siebertz. A survey on the parameterized complexity of the independent set and (connected) dominating set reconfiguration problems. arXiv preprint, 2022. URL:
  6. Hajo Broersma and Xueliang Li. Spanning trees with many or few colors in edge-colored graphs. Discuss. Math. Graph Theory, 17(2):259-269, 1997. Google Scholar
  7. Lily Chen, Xueliang Li, and Yongtang Shi. The complexity of determining the rainbow vertex-connection of a graph. Theoretical Computer Science, 412(35):4531-4535, 2011. Google Scholar
  8. Bruno Courcelle and Mohamed Mosbah. Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science, 109(1-2):49-82, 1993. Google Scholar
  9. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  10. Marzio De Biasi and Juho Lauri. On the complexity of restoring corrupted colorings. Journal of Combinatorial Optimization, 37(4):1150-1169, 2019. Google Scholar
  11. Rod G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness II: On completeness for W[1]. Theoretical Computer Science, 141(1-2):109-131, 1995. Google Scholar
  12. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. URL:
  13. Michael R. Fellows, Mario Grobler, Nicole Megow, Amer E. Mouawad, Vijayaragunathan Ramamoorthi, Frances A. Rosamond, Daniel Schmand, and Sebastian Siebertz. On solution discovery via reconfiguration. In ECAI 2023 - 26th European Conference on Artificial Intelligence, volume 372 of Frontiers in Artificial Intelligence and Applications, pages 700-707. IOS Press, 2023. Google Scholar
  14. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. Google Scholar
  15. Michael R. Garey, David S. Johnson, and R Endre Tarjan. The planar hamiltonian circuit problem is NP-complete. SIAM Journal on Computing, 5(4):704-714, 1976. Google Scholar
  16. Valentin Garnero, Konstanty Junosza-Szaniawski, Mathieu Liedloff, Pedro Montealegre, and Paweł Rzazewski. Fixing improper colorings of graphs. Theoretical Computer Science, 711:66-78, 2018. Google Scholar
  17. Guilherme Gomes, Sérgio H. Nogueira, and Vinicius F. dos Santos. Some results on vertex separator reconfiguration. arXiv preprint, 2020. URL:
  18. Guilherme C. M. Gomes, Clément Legrand-Duchesne, Reem Mahmoud, Amer E. Mouawad, Yoshio Okamoto, Vinícius Fernandes dos Santos, and Tom C. van der Zanden. Minimum separator reconfiguration. In 18th International Symposium on Parameterized and Exact Computation, IPEC 2023, volume 285 of LIPIcs, pages 9:1-9:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. Google Scholar
  19. Sushmita Gupta, Sanjukta Roy, Saket Saurabh, and Meirav Zehavi. Parameterized algorithms and kernels for rainbow matching. Algorithmica, 81(4):1684-1698, 2019. Google Scholar
  20. Jan van den Heuvel. The complexity of change. Surveys in Combinatorics, 409(2013):127-160, 2013. Google Scholar
  21. Takehiro Ito, Erik D. Demaine, Nicholas J.A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theoretical Computer Science, 412(12-14):1054-1065, 2011. Google Scholar
  22. Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. Shortest reconfiguration of perfect matchings via alternating cycles. SIAM Journal on Discrete Mathematics, 36(2):1102-1123, 2022. Google Scholar
  23. Marcin Kamiński, Paul Medvedev, and Martin Milanič. Shortest paths between shortest paths. Theoretical Computer Science, 412(39):5205-5210, 2011. Google Scholar
  24. Marcin Kaminski, Paul Medvedev, and Martin Milanic. Complexity of independent set reconfigurability problems. Theoretical Computer Science, 439:9-15, 2012. Google Scholar
  25. Van Bang Le and Florian Pfender. Complexity results for rainbow matchings. Theoretical Computer Science, 524:27-33, 2014. URL:
  26. Nicolas El Maalouly and Raphael Steiner. Exact matching in graphs of bounded independence number. In 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022, volume 241 of LIPIcs, pages 46:1-46:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  27. Dániel Marx, Barry O'Sullivan, and Igor Razgon. Finding small separators in linear time via treewidth reduction. ACM Transactions on Algorithms (TALG), 9(4):1-35, 2013. Google Scholar
  28. Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. Google Scholar
  29. Christos Nomikos, Aris Pagourtzis, and Stathis Zachos. Randomized and approximation algorithms for blue-red matching. In 32nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2022, pages 715-725. Springer, 2007. Google Scholar
  30. Christos H. Papadimitriou and Mihalis Yannakakis. The complexity of restricted spanning tree problems. Journal of the ACM (JACM), 29(2):285-309, 1982. Google Scholar
  31. Kei Uchizawa, Takanori Aoki, Takehiro Ito, Akira Suzuki, and Xiao Zhou. On the rainbow connectivity of graphs: complexity and fpt algorithms. Algorithmica, 67:161-179, 2013. Google Scholar