Advertisement

Using contracted solution graphs for solving reconfiguration problems

  • 79 Accesses

Abstract

We introduce in a general setting a dynamic programming method for solving reconfiguration problems. Our method is based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. Our general framework captures the approach behind known reconfiguration results of Bonsma (Discrete Appl Math 231:95–112, 2017) and Hatanaka et al. (IEICE Trans Fundam Electron Commun Comput Sci 98(6):1168–1178, 2015). As a third example, we apply the method to the following well-studied problem: given two k-colorings \(\alpha \) and \(\beta \) of a graph G, can \(\alpha \) be modified into \(\beta \) by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? This problem is known to be PSPACE-hard even for bipartite planar graphs and \(k=4\). By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for \((k-2)\)-connected chordal graphs.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    In [8], it is first shown that the list-coloring version of the \(\mathcal {C}_k\)-Reachability problem is PSPACE-complete for connected planar bipartite 4-colorable graphs. In the list-coloring version every vertex is assigned a list of two or three allowed colors. In the next step of the transformation, to the \(\mathcal {C}_k\)-Reachability problem, the list-coloring constraints are simulated by making every vertex adjacent to appropriately colored vertices of a small bipartite graph with a frozen 4-coloring. A frozen k-coloring is a k-coloring where every vertex is adjacent to vertices of each other color, so no vertex can be recolored. Since we do not care about planarity, we can instead simply take one highly connected bipartite graph with a frozen k-coloring and link all vertices of the first graph to the appropriate vertices of this new graph, to enforce the list-color constraints, while maintaining bipartiteness and ensuring \((k-2)\)-connectedness.

  2. 2.

    Since node names are irrelevant, we will simply write \((H,\ell )=\mathcal {S}^c(G,T)\) to denote that there is a label preserving isomorphism between the two. More formally, \(\mathcal {S}^c(G,T)\) can be seen as a class of labeled graphs that are equivalent under labeled isomorphisms.

  3. 3.

    It is possible for the given example to choose two solutions \(\alpha \) and \(\beta \), and correctly mark an \(\alpha \)-node x with respect to a one certificate \(S^1\), and a \(\beta \)-node y with respect to another certificate \(S^2\), such that \(\alpha \) and \(\beta \) are in different components of \(\mathcal {S}(G)\), but x and y are in the same component of \(\mathcal {S}^c(G,T)\). This is clearly not desirable; see Lemma 2.

References

  1. 1.

    Bodlaender, H., Bonsma, P., Lokshtanov, D.: The fine details of fast dynamic programming over tree decompositions. In: Proceedings of IPEC. Lecture Notes in Computer Science, vol. 8246, pp. 41–53. Springer (2013)

  2. 2.

    Bonamy, M., Bousquet, N.: Recoloring graphs via tree decompositions. Eur. J. Comb. 69, 200–213 (2018)

  3. 3.

    Bonamy, M., Dabrowski, K.K., Feghali, C., Johnson, M., Paulusma, D.: Recognizing graphs close to bipartite graphs with an application to colouring reconfiguration. CoRR, arXiv:1707.09817 (2017)

  4. 4.

    Bonamy, M., Johnson, M., Lignos, I., Patel, V., Paulusma, D.: Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs. J. Comb. Optim. 27, 132–143 (2014)

  5. 5.

    Bonsma, P.: The complexity of rerouting shortest paths. Theor. Comput. Sci. 510, 1–12 (2013)

  6. 6.

    Bonsma, P.: Independent set reconfiguration in cographs and their generalizations. J. Graph Theory 83(2), 164–195 (2016)

  7. 7.

    Bonsma, P.: Rerouting shortest paths in planar graphs. Discrete Appl. Math. 231, 95–112 (2017)

  8. 8.

    Bonsma, P., Cereceda, L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theor. Comput. Sci. 410(50), 5215–5226 (2009)

  9. 9.

    Bonsma, P., Kamiński, M., Wrochna, M.: Reconfiguring independent sets in claw-free graphs. In: Proceedings of SWAT 2014. Lecture Notes in Computer Science, vol. 8503, pp. 86–97. Springer (2014)

  10. 10.

    Bonsma, P., Mouawad, A., Nishimura, N., Raman, V.: The complexity of bounded length graph recoloring and CSP reconfiguration. In: Proceedings of IPEC 2014. Lecture Notes in Computer Science, vol. 8894, pp. 110–121. Springer (2014)

  11. 11.

    Bonsma, P., Paulusma, D.: Using contracted solution graphs for solving reconfiguration problems. In: Proceedings of MFCS 2016. LIPIcs, vol. 58, pp. 20:1–20:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)

  12. 12.

    Brandstädt, A., Le, V., Spinrad, J.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)

  13. 13.

    Cereceda, L., van den Heuvel, J., Johnson, M.: Connectedness of the graph of vertex-colourings. Discrete Math. 308(5), 913–919 (2008)

  14. 14.

    Cereceda, L., van den Heuvel, J., Johnson, M.: Mixing 3-colourings in bipartite graphs. Eur. J. Comb. 30(7), 1593–1606 (2009)

  15. 15.

    Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between 3-colorings. J. Graph Theory 67(1), 69–82 (2011)

  16. 16.

    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Berlin (2010)

  17. 17.

    Feghali, C., Johnson, M., Paulusma, D.: A reconfigurations analogue of Brooks’ theorem and its consequences. J. Graph Theory 83(4), 340–358 (2016)

  18. 18.

    Gopalan, P., Kolaitis, P.G., Maneva, E.N., Papadimitriou, C.H.: The connectivity of boolean satisfiability: computational and structural dichotomies. SIAM J. Comput. 38(6), 2330–2355 (2009)

  19. 19.

    Haddadan, A., Ito, T., Mouawad, A.E., Nishimura, N., Ono, H., Suzuki, A., Tebbal, Y.: The complexity of dominating set reconfiguration. Theor. Comput. Sci. 651, 37–49 (2016)

  20. 20.

    Hatanaka, T., Ito, T., Zhou, X.: The list coloring reconfiguration problem for bounded pathwidth graphs. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 98(6), 1168–1178 (2015)

  21. 21.

    Hatanaka, T., Ito, T., Zhou, X.: The coloring reconfiguration problem on specific graph classes. In: Proceedings of COCOA 2017. Lecture Notes in Computer Science, vol. 10627, pp. 152–162 (2017)

  22. 22.

    Hatanaka, T., Ito, T., Zhou, X.: Parameterized complexity of the list coloring reconfiguration problem with graph parameters. Theor. Comput. Sci. 739, 65–79 (2018)

  23. 23.

    Ito, T., Demaine, E., Harvey, N., Papadimitriou, C., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theor. Comput. Sci. 412(12–14), 1054–1065 (2011)

  24. 24.

    Ito, T., Demaine, E.D.: Approximability of the subset sum reconfiguration problem. J. Comb. Optim. 28(3), 639–654 (2014)

  25. 25.

    Ito, T., Kamiński, M., Demaine, E.: Reconfiguration of list edge-colorings in a graph. Discrete Appl. Math. 160(15), 2199–2207 (2012)

  26. 26.

    Ito, T., Kawamura, K., Ono, H., Zhou, X.: Reconfiguration of list \({L} (2, 1)\)-labelings in a graph. Theor. Comput. Sci. 544, 84–97 (2014)

  27. 27.

    Ito, T., Kawamura, K., Zhou, X.: An improved sufficient condition for reconfiguration of list edge-colorings in a tree. IEICE Trans. Inf. Syst. 95(3), 737–745 (2012)

  28. 28.

    Johnson, M., Kratsch, D., Kratsch, S., Patel, V., Paulusma, D.: Finding shortest paths between graph colourings. Algorithmica 75(2), 295–321 (2016)

  29. 29.

    Kamiński, M., Medvedev, P., Milanič, M.: Complexity of independent set reconfigurability problems. Theor. Comput. Sci. 439, 9–15 (2012)

  30. 30.

    Kloks, T.: Treewidth: Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer, Berlin (1994)

  31. 31.

    König, F.G., Lübbecke, M.E., Möhring, R.H., Schäfer, G., Spenke, I.: Solutions to real-world instances of PSPACE-complete stacking. In: Proceedings of ESA 2007. Lecture Notes in Computer Science, vol. 4698, pp. 729–740 (2007)

  32. 32.

    Lokshtanov, D., Mouawad, A.E., Panolan, F., Ramanujan, M.S., Saurabh, S.: Reconfiguration on sparse graphs. J. Comput. Syst. Sci. 95, 122–131 (2018)

  33. 33.

    Mouawad, A.E., Nishimura, N., Pathak, V., Raman, V.: Shortest reconfiguration paths in the solution space of Boolean formulas. SIAM J. Discrete Math. 31(3), 2185–2200 (2017)

  34. 34.

    Mouawad, A.E., Nishimura, N., Raman, V., Siebertz, S.: Vertex cover reconfiguration and beyond. Algorithms 11(2), 20 (2018)

  35. 35.

    Mouawad, A.E., Nishimura, N., Raman, V., Simjour, N., Suzuki, A.: On the parameterized complexity of reconfiguration problems. Algorithmica 78(1), 274–297 (2017)

  36. 36.

    Mouawad, A.E., Nishimura, N., Raman, V., Wrochna, M.: Reconfiguration over tree decompositions. In: Proceedings of IPEC 2014. Lecture Notes in Computer Science, vol. 8894, pp. 246–257. Springer (2014)

  37. 37.

    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)

  38. 38.

    Nishimura, N.: Introduction to reconfiguration. Algorithms 11(4), 52 (2018)

  39. 39.

    Roberts, F.S.: Indifference graphs. Proof Techniques in Graph Theory, pp. 139–146. Academic Press, New York (1969)

  40. 40.

    van den Heuvel, J.: The complexity of change. Surv. Comb. 409, 127–160 (2013)

  41. 41.

    van der Zanden, T.: Parameterized complexity of graph constraint logic. In: Proceedings of IPEC 2015. LIPIcs, vol. 43, pp. 282–293. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015)

  42. 42.

    Wrochna, M.: Homomorphism reconfiguration via homotopy. In: Proceedings of STACS 2015. LIPIcs, vol. 30, pp. 730–742. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015)

  43. 43.

    Wrochna, M.: Reconfiguration in bounded bandwidth and tree-depth. J. Comput. Syst. Sci. 93, 1–10 (2018)

Download references

Acknowledgements

The authors would like to thank Carl Feghali and Matthew Johnson for fruitful discussions and two anonymous reviewers for helpful comments.

Author information

Correspondence to Daniël Paulusma.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

An extended abstract of this paper appeared in the proceedings of MFCS 2016 [11].

Paul Bonsma: Supported by the European Community’s Seventh Framework Programme (FP7/2007-2013), Grant agreement No. 317662. Daniël Paulusma: Supported by EPSRC Grant EP/K025090/1.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bonsma, P., Paulusma, D. Using contracted solution graphs for solving reconfiguration problems. Acta Informatica 56, 619–648 (2019). https://doi.org/10.1007/s00236-019-00336-8

Download citation