What Makes Equitable Connected Partition Easy
We study the Equitable Connected Partition problem: partitioning the vertices of a graph into a specified number of classes, such that each class of the partition induces a connected subgraph, so that the classes have cardinalities that differ by at most one. We examine the problem from the parameterized complexity perspective with respect to various (aggregate) parameterizations involving such secondary measurements as: (1) the number of partition classes, (2) the treewidth, (3) the pathwidth, (4) the minimum size of a feedback vertex set, (5) the minimum size of a vertex cover, (6) and the maximum number of leaves in a spanning tree of the graph. In particular, we show that the problem is W-hard with respect to the first four combined, while it is fixed-parameter tractable with respect to each of the last two alone. The hardness result holds even for planar graphs. The problem is in XP when parameterized by treewidth, by standard dynamic programming techniques. Furthermore, we show that the closely related problem of Equitable Coloring (equitably partitioning the vertices into a specified number of independent sets) is FPT parameterized by the maximum number of leaves in a spanning tree of the graph.
Unable to display preview. Download preview PDF.
- 1.Altman, M.: Is automation the answer? the computational complexity of automated redistricting. Rutgers Computer and Technology Law Journal 23, 81–142 (2007)Google Scholar
- 3.Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
- 5.Estivill-Castro, V., Fellows, M.R., Langston, M.A., Rosamond, F.A.: FPT is P-Time Extremal Structure I. In: Hajo Broersma, M.J., Szeider, S. (eds.) ACiD. Texts in Algorithmics, vol. 4, pp. 1–41. King’s College, London (2005)Google Scholar
- 6.Fellows, M.R., Fomin, F.V., Lokshtanov, D., Rosamond, F.A., Saurabh, S., Szeider, S., Thomassen, C.: On the complexity of some colorful problems parameterized by treewidth. In: Dress, A.W.M., Xu, Y., Zhu, B. (eds.) COCOA. LNCS, vol. 4616, pp. 366–377. Springer, Heidelberg (2007)CrossRefGoogle Scholar
- 8.Fiala, J., Golovach, P.A., Kratochvíl, J.: Parameterized complexity of coloring problems: Treewidth versus vertex cover. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 221–230. Springer, Heidelberg (2009)Google Scholar
- 12.Ito, T., Zhou, X., Nishizeki, T.: Partitioning a weighted graph to connected subgraphs of almost uniform size. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 365–376. Springer, Heidelberg (2004)Google Scholar