Abstract
Let G be a graph such that each edge has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. Suppose that we are given two list edge-colorings \(f_0\) and \(f_r\) of G, and asked whether there exists a sequence of list edge-colorings of G between \(f_0\) and \(f_r\) such that each list edge-coloring can be obtained from the previous one by changing a color assignment of exactly one edge. This problem is known to be PSPACE-complete for every integer \(k \ge 6\) and planar graphs of maximum degree three, but any computational hardness was unknown for the non-list variant in which every edge has the same list of k colors. In this paper, we first improve the known result by proving that, for every integer \(k \ge 4\), the problem remains PSPACE-complete even for planar graphs of maximum degree three and bounded bandwidth. Since the problem is known to be solvable in polynomial time if \(k \le 3\), our result gives a sharp analysis of the complexity status with respect to the number k of colors. We then give the first computational hardness result for the non-list variant: for every integer \(k \ge 5\), the non-list variant is PSPACE-complete even for planar graphs of maximum degree k and bandwidth linear in k.
This work is partially supported by JSPS KAKENHI Grant Numbers JP26730001 (A. Suzuki), JP15H00849 and JP16K00004 (T. Ito), and JP16K00003 (X. Zhou).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bonamy, M., Bousquet, N.: Recoloring bounded treewidth graphs. Electron. Notes Discret. Math. 44, 257–262 (2013)
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)
Bonsma, P., Cereceda, L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoret. Comput. Sci. 410, 5215–5226 (2009)
Bonsma, P., Mouawad, A.E., Nishimura, N., Raman, V.: The complexity of bounded length graph recoloring and CSP reconfiguration. In: Cygan, M., Heggernes, P. (eds.) IPEC 2014. LNCS, vol. 8894, pp. 110–121. Springer, Heidelberg (2014). doi:10.1007/978-3-319-13524-3_10
Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between 3-colorings. J. Graph Theory 67, 69–82 (2011)
Demaine, E.D., Demaine, M.L., Fox-Epstein, E., Hoang, D.A., Ito, T., Ono, H., Otachi, Y., Uehara, R., Yamada, T.: Linear-time algorithm for sliding tokens on trees. Theoret. Comput. Sci. 600, 132–142 (2015)
Hatanaka, T., Ito, T., Zhou, X.: The list coloring reconfiguration problem for bounded pathwidth graphs. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E98-A, 1168–1178 (2015)
Hearn, R.A., Demaine, E.D.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theoret. Comput. Sci. 343, 72–96 (2005)
van den Heuvel, J.: The Complexity of Change. Surveys in Combinatorics 2013. London Mathematical Society Lecture Notes Series 409 (2013)
Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theoret. Comput. Sci. 412, 1054–1065 (2011)
Ito, T., Kamiński, M., Demaine, E.D.: Reconfiguration of list edge-colorings in a graph. Discret. Appl. Math. 160, 2199–2207 (2012)
Ito, T., Kawamura, K., Zhou, X.: An improved sufficient condition for reconfiguration of list edge-colorings in a tree. IEICE Trans. Inf. Syst. E95-D, 737–745 (2012)
Ito, T., Ono, H., Otachi, Y.: Reconfiguration of cliques in a graph. In: Jain, R., Jain, S., Stephan, F. (eds.) TAMC 2015. LNCS, vol. 9076, pp. 212–223. Springer, Heidelberg (2015). doi:10.1007/978-3-319-17142-5_19
Johnson, M., Kratsch, D., Kratsch, S., Patel, V., Paulusma, D.: Finding shortest paths between graph colourings. Algorithmica 75, 295–321 (2016)
Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. J. Comput. Syst. Sci. 4, 177–192 (1970)
Wrochna, M.: Reconfiguration in bounded bandwidth and treedepth (2014). arXiv:1405.0847
van der Zanden, T.C.: Parameterized complexity of graph constraint logic. In: Proceedings of IPEC 2015, LIPIcs, vol. 9076, pp. 282–293 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Osawa, H., Suzuki, A., Ito, T., Zhou, X. (2017). The Complexity of (List) Edge-Coloring Reconfiguration Problem. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_27
Download citation
DOI: https://doi.org/10.1007/978-3-319-53925-6_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-53924-9
Online ISBN: 978-3-319-53925-6
eBook Packages: Computer ScienceComputer Science (R0)