Abstract
Given a directed graph on n vertices and an integer parameter k, the feedback vertex (arc) set problem asks whether the given graph has a set of k vertices (arcs) whose removal results in an acyclic directed graph. The parameterized complexity of these problems, in the framework introduced by Downey and Fellows, is a long standing open problem in the area. We address these problems in the well studied class of directed graphs called tournaments.
While the feedback vertex set problem is easily seen to be fixed parameter tractable in tournaments, we show that the feedback arc set problem is also fixed parameter tractable. Then we address the parametric dual problems (where the k is replaced by ‘all but k’ in the questions) and show that they are fixed parameter tractable in oriented directed graphs (where there is at most one directed arc between a pair of vertices). More specifically, the dual problem we show fixed parameter tractable are: Given an oriented directed graph, is there a subset of k vertices (arcs) that forms an acyclic directed subgraph of the graph?
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arvind, V., Fellows, M.R., Mahajan, M., Raman, V., Rao, S.S., Rosamond, F.A., Subramanian, C.R.: Parametric Duality and Fixed Parameter Tractability (2001) (manuscript)
Bang-Jensen, J., Gutin, G.: Digraphs Theory, Algorithms and Applications. Springer, Heidelberg (2001)
Downey, R., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1998)
Even, G. (Seffi) Naor, J., Schieber, B., Sudan, M.: Approximating Minimum Feedback Sets and Multicuts in Directed Graphs. Algorithmica 20, 151–174 (1998)
Fellows, M., Hallett, M., Korostensky, C., Stege, U.: Analogs and Duals of the MAST Problem for Sequences and Trees. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 103–114. Springer, Heidelberg (1998)
Itai, A., Rodeh, M.: Finding a Minimum Circuit in a Graph. Siam Journal of Computing 7(4), 413–423 (1978)
Khot, S., Raman, V.: Parameterized Complexity of Finding Subgraphs with Hereditary Properties. Theoretical Computer Science 289, 997–1008 (2002)
Mahajan, M., Raman, V.: Parameterizing above Guaranteed Values: MaxSat and MaxCut. Journal of Algorithms 31, 335–354 (1999)
Niedermeier, R., Rossmanith, P.: An efficient Fixed Parameter Algorithm for 3-Hitting Set. Journal of Discrete Algorithms 2(1) (2001)
Poljak, S., Turzik, D.: A Polynomial Algorithm for Constructing a Large Bipartite Subgraph, with an Application to a Satisfiability Problem. Canad. J. Math. 34(3), 519–524 (1982)
Raman, V., Saurabh, S., Subramanian, C.R.: Faster Fixed Parameter Tractable Algorithms for Undirected Feedback Vertex Set’. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 241–248. Springer, Heidelberg (2002)
Speckenmeyer, E.: On Feedback Problems in Digraphs. In: Nagl, M. (ed.) WG 1989. LNCS, vol. 411, pp. 218–231. Springer, Heidelberg (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Raman, V., Saurabh, S. (2003). Parameterized Complexity of Directed Feedback Set Problems in Tournaments. In: Dehne, F., Sack, JR., Smid, M. (eds) Algorithms and Data Structures. WADS 2003. Lecture Notes in Computer Science, vol 2748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45078-8_42
Download citation
DOI: https://doi.org/10.1007/978-3-540-45078-8_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40545-0
Online ISBN: 978-3-540-45078-8
eBook Packages: Springer Book Archive