A Complexity Dichotomy for Finding Disjoint Solutions of Vertex Deletion Problems
We investigate the computational complexity of a general “compression task” centrally occurring in the recently developed technique of iterative compression for exactly solving NP-hard minimization problems. The core issue (particularly but not only motivated by iterative compression) is to determine the computational complexity of, given an already inclusion-minimal solution for an underlying (typically NP-hard) vertex deletion problem in graphs, to find a better disjoint solution. The complexity of this task is so far lacking a systematic study. We consider a large class of vertex deletion problems on undirected graphs and show that, except for few cases which are polynomial-time solvable, the others are NP-complete. This class includes problems such as Vertex Cover (here the corresponding compression task is decidable in polynomial time) or Undirected Feedback Vertex Set (here the corresponding compression task is NP-complete).
KeywordsVertex Cover Graph Property Complexity Dichotomy Induce Subgraph Vertex Deletion
Unable to display preview. Download preview PDF.
- 7.Greenwell, D.L., Hemminger, R.L., Klerlein, J.B.: Forbidden subgraphs. In: Proc. 4th CGTC, pp. 389–394 (1973)Google Scholar
- 9.Guo, J., Moser, H., Niedermeier, R.: Iterative compression for exactly solving NP-hard minimization problems. In: Algorithmics of Large and Complex Networks. LNCS, vol. 5515, pp. 65–80. Springer, Heidelberg (2009)Google Scholar
- 10.Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst. (2009); available electronicallyGoogle Scholar
- 13.Marx, D.: Chordal deletion is fixed-parameter tractable. Algorithmica (2009); available electronicallyGoogle Scholar
- 16.Razgon, I., O’Sullivan, B.: Almost 2-SAT is fixed-parameter tractable. J. Comput. System Sci. (2009); available electronicallyGoogle Scholar