On Digraph Width Measures in Parameterized Algorithmics
In contrast to undirected width measures (such as tree-width or clique-width), which have provided many important algorithmic applications, analogous measures for digraphs such as DAG-width or Kelly-width do not seem so successful. Several recent papers, e.g. those of Kreutzer–Ordyniak, Dankelmann–Gutin–Kim, or Lampis–Kaouri–Mitsou, have given some evidence for this. We support this direction by showing that many quite different problems remain hard even on graph classes that are restricted very beyond simply having small DAG-width. To this end, we introduce new measures K-width and DAG-depth. On the positive side, we also note that taking Kanté’s directed generalization of rank-width as a parameter makes many problems fixed parameter tractable.
Unable to display preview. Download preview PDF.
- [BG04]Berwanger, D., Grädel, E.: Entanglement – a measure for the complexity of directed graphs with applications to logic and games. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 209–223. Springer, Heidelberg (2005)Google Scholar
- [BJK09]Bang-Jensen, J., Kriesell, M.: Disjoint directed and undirected paths and cycles in digraphs. Technical Report PP-2009-03, University of South Denmark (2009)Google Scholar
- [BXTV08]Bui-Xuan, B.-M., Telle, J., Vatshelle, M.: H-join and algorithms on graphs of bounded rank-width (submitted) (November 2008)Google Scholar
- [DGK08]Dankelmann, P., Gutin, G., Kim, E.: On complexity of minimun leaf out-branching. arXiv:0808.0980v1 (August 2008)Google Scholar
- [FGLS09]Fomin, F., Golovach, P., Lokshtanov, D., Saurab, S.: Clique-width: On the price of generality. In: SODA 2009, pp. 825–834. SIAM, Philadelphia (2009)Google Scholar
- [GH08]Ganian, R., Hliněný, P.: Automata approach to graphs of bounded rank-width. In: IWOCA 2008, pp. 4–15 (2008)Google Scholar
- [GH09]Ganian, R., Hliněný, P.: Better polynomial algorithms on graphs of bounded rank-width. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 266–277. Springer, Heidelberg (2009)Google Scholar
- [Gru08]Gruber, H.: Digraph complexity measures and applications in formal language theory. In: MEMICS 2008, pp. 60–67 (2008)Google Scholar
- [HO06]Hliněný, P., Obdržálek, J.: Escape-width: Measuring ”width” of digraphs. Presented at Sixth Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications (2006)Google Scholar
- [Kan08]Kanté, M.: The rank-width of directed graphs. arXiv:0709.1433v3 (March 2008)Google Scholar
- [NdM06]Nešetřil, J., Ossona de Mendez, P.: Tree-depth, subgraph coloring and homomorphism bounds. European J. Combin. 27(6), 1024–1041 (2006)Google Scholar
- [Obd03]Obdržálek, J.: Fast mu-calculus model checking when tree-width is bounded. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 80–92. Springer, Heidelberg (2003)Google Scholar
- [vL76]van Leeuwen, J.: Having a Grundy-numbering is NP-complete. Technical Report 207, The Pennsylvania State University (September 1976)Google Scholar