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A Complexity Dichotomy for Finding Disjoint Solutions of Vertex Deletion Problems

Published:01 March 2011Publication History
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Abstract

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 the following task: given an already inclusion-minimal solution for an underlying (typically NP-hard) vertex deletion problem in graphs, find a smaller 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 a few cases are polynomial-time solvable, and the others are NP-hard. The considered class of vertex deletion problems includes Vertex Cover (where the compression task is polynomial time) and Undirected Feedback Vertex Set (where the compression task is NP-complete).

References

  1. Aarts, E. and Lenstra, J. K. 1997. Local Search in Combinatorial Optimization. Wiley. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Abello, J., Resende, M. G. C., and Sudarsky, S. 2002. Massive quasi-clique detection. In Proceedings of the 5th Latin American Symposium on Theoretical Informatics (LATIN ’02). Lecture Notes in Computer Science, vol. 2286. Springer, 598--612. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Böckenhauer, H.-J., Hromkovič, J., Mömke, T., and Widmayer, P. 2008. On the hardness of reoptimization. In Proceedings of the 34th Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM’08). Lecture Notes in Computer Science, vol. 4910. Springer, 50--65. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Cai, L. 1996. Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58, 4, 171--176. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Chen, J., Fomin, F. V., Liu, Y., Lu, S., and Villanger, Y. 2008a. Improved algorithms for feedback vertex set problems. J. Comput. Syst. Sci. 74, 7, 1188--1198. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Chen, J., Liu, Y., Lu, S., O’Sullivan, B., and Razgon, I. 2008b. A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55, 5. Article 21. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Dehne, F. K. H. A., Fellows, M. R., Langston, M. A., Rosamond, F. A., and Stevens, K. 2007. An O (2O(k)n 3) FPT algorithm for the undirected feedback vertex set problem. Theory Comput. Syst. 41, 3, 479--492. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Gabow, H. N. and Tarjan, R. E. 1989. Faster scaling algorithms for network problems. SIAM J. Comput. 18, 5, 1013--1036. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Garey, M. R. and Johnson, D. S. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Greenwell, D. L., Hemminger, R. L., and Klerlein, J. B. 1973. Forbidden subgraphs. In Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing. 389--394.Google ScholarGoogle Scholar
  11. Guo, J. 2006. Algorithm design techniques for parameterized graph modification problems. Ph.D. thesis, Institut für Informatik, Friedrich-Schiller-Universität Jena.Google ScholarGoogle Scholar
  12. Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., and Wernicke, S. 2006. Compression-Based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Comput. Syst. Sci. 72, 8, 1386--1396. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Guo, J., Moser, H., and Niedermeier, R. 2009. Iterative compression for exactly solving NP-hard minimization problems. In Algorithmics of Large and Complex Networks. Lecture Notes in Computer Science, vol. 5515. Springer, 65--80. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. ffner, F., Komusiewicz, C., Moser, H., and Niedermeier, R. 2010. Fixed-Parameter algorithms for cluster vertex deletion. Theory Comput. Syst. 47, 1, 196--217. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Khot, S. and Raman, V. 2002. Parameterized complexity of finding subgraphs with hereditary properties. Theor. Comput. Sci. 289, 2, 997--1008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Krawczyk, A. 1999. The complexity of finding a second Hamiltonian cycle in cubic graphs. J. Comput. Syst. Sci. 58, 3, 641--647. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Lewis, J. M. and Yannakakis, M. 1980. The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20, 2, 219--230.Google ScholarGoogle ScholarCross RefCross Ref
  18. Marx, D. 2010. Chordal deletion is fixed-parameter tractable. Algorithmica 57, 4, 747--768. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Marx, D. and Schlotter, I. 2007. Obtaining a planar graph by vertex deletion. In Proceedings of the 33rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG’07). Lecture Notes in Computer Science, vol. 4769. Springer, 292--303. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Moser, H., Niedermeier, R., and Sorge, M. 2009. Algorithms and experiments for clique relaxations---Finding maximum s-plexes. In Proceedings of the 8th International Symposium on Experimental Algorithms (SEA’09). Lecture Notes in Computer Science, vol. 5526. Springer, 233--244. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Niedermeier, R. 2006. Invitation to Fixed-Parameter Algorithms. Oxford University Press.Google ScholarGoogle Scholar
  22. Nishimura, N., Ragde, P., and Thilikos, D. M. 2005. Fast fixed-parameter tractable algorithms for nontrivial generalizations of Vertex Cover. Discr. Appl. Math. 152, 1--3, 229--245. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Papadimitriou, C. H. 1994. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48, 3, 498--532. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Razgon, I. and O’Sullivan, B. 2009. Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci. 75, 8, 435--450. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Reed, B., Smith, K., and Vetta, A. 2004. Finding odd cycle transversals. Oper. Res. Lett. 32, 4, 299--301. Google ScholarGoogle ScholarDigital LibraryDigital Library

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