A cubic Kernel for feedback vertex set and loop cutset

Theory of Computing Systems

Volumn 46, Issue 3, 2010, Pages 566-597

  Bodlaender, Hans L ^a   van Dijk, Thomas C ^a  

^a UTRECHT UNIVERSITY   (Netherlands)

Author keywords

Algorithms; Data reduction; Feedback vertex set; Fixed parameter tractability; Graphs; Kernelization algorithms; Loop cutset; Polynomial kernels; Preprocessing

Indexed keywords

CUT-SET; FEEDBACK VERTEX SET; FIXED-PARAMETER TRACTABILITY; GRAPHS; KERNELIZATION; POLYNOMIAL KERNELS; PREPROCESSING;

ALGORITHMS; DATA REDUCTION; POLYNOMIAL APPROXIMATION;

GRAPH THEORY;

EID: 77951258368     PISSN: 14324350     EISSN: 14330490     Source Type: Journal    
DOI: 10.1007/s00224-009-9234-2     Document Type: Article

References (40)

1. Abu-Khzam, F. N., Collins, R. L., Fellows, M. R., Langston, M. A., Suters, W. H., Symons, C. T.: Kernelization algorithms for the vertex cover problem: theory and experiments. In: Proc. 6th ACM-SIAM ALENEX, 2004, pp. 62-69. ACM-SIAM, New York, Philadelphia (2004).
2. Alber, J., Fellows, M. R., Niedermeier, R.: Polynomial-time data reduction for dominating sets. J. ACM 51, 363-384 (2004).
3. Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math. 12, 289-297 (1999).
4. Becker, A., Bar-Yehuda, R., Geiger, D.: Randomized algorithms for the loop cutset problem. J. Artif. Intell. Res. 12, 219-234 (2000).
5. Becker, A., Geiger, D.: Optimization of Pearl's method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artif. Intell. 83, 167-188 (1996).
6. Bodlaender, H. L.: On disjoint cycles. Int. J. Found. Comput. Sci. 5(1), 59-68 (1994).
7. Bodlaender, H. L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305-1317 (1996).
8. Bodlaender, H. L.: Necessary edges in k-chordalizations of graphs. J. Comb. Optim. 7, 283-290 (2003).
9. Bodlaender, H. L., Penninkx, E.: A linear kernel for planar feedback vertex set. In: Grohe, M., Niedermeier, R. (eds.) Proceedings 3rd International Workshop on Parameterized and Exact Computation, IWPEC 2008. Lecture Notes in Computer Science, vol. 5018, pp. 160-171. Springer, Berlin (2008).
10. Bodlaender, H. L., Downey, R. G., Fellows, M. R., Hermelin, D.: On problems without polynomial kernels (extended abstract). In: Proceedings 35th International Colloquium on Automata, Languages and Programming, ICALP 2008. Lecture Notes in Computer Science, vol. 5125, pp. 563-574. Springer, Berlin (2008).
11. Burrage, K., Estivill-Castro, V., Fellows, M. R., Langston, M. A., Mac, S., Rosamond, F. A.: The undirected feedback vertex set problem has a poly(k) kernel. In: Bodlaender, H. L., Langston, M. A. (eds.) Proceedings 2nd International Workshop on Parameterized and Exact Computation, IWPEC 2006. Lecture Notes in Computer Science, vol. 4169, pp. 192-202. Springer, Berlin (2006).
12. Chen, J., Kanj, I. A., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41, 280-301 (2001).
13. Chen, J., Liu, Y., Lu, S., O'Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. In: STOC '08: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, New York, USA, 2008, pp. 177-186. ACM, New York (2008).
14. Clautiaux, F., Carlier, J., Moukrim, A., Négre, S.: New lower and upper bounds for graph treewidth. In: Rolim, J. D. P. (ed.) Proceedings International Workshop on Experimental and Efficient Algorithms, WEA 2003. Lecture Notes in Computer Science, vol. 2647, pp. 70-80. Springer, Berlin (2003).
15. 3) FPT algorithm for the undirected feedback vertex set problem. In: Wang, L. (ed.) Proceedings 11th International Computing and Combinatorics Conference COCOON 2005. Lecture Notes in Computer Science, vol. 3595, pp. 859-869. Springer, Berlin (2005).
16. Downey, R. G., Fellows, M. R.: Fixed-parameter tractability and completeness. Congr. Numer. 87, 161-178 (1992).
17. Downey, R. G., Fellows, M. R.: Parameterized Complexity. Springer, Berlin (1998).
18. Festa, P., Pardalos, P. M., Resende, M. G. C.: Feedback set problems. In: Handbook of Combinatorial Optimization, vol. A, pp. 209-258. Kluwer, Amsterdam (1999).
19. n). In: Bodlaender, H. L., Langston, M. A. (eds.) Proceedings 2nd International Workshop on Parameterized and Exact Computation, IWPEC 2006. Lecture Notes in Computer Science, vol. 4169, pp. 183-191. Springer, Berlin (2006).
20. Gerards, A. M. H.: Matching. In: Ball, M. O. et al. (ed.) Network Models. Handbooks in Operations Research and Management Sciences, vol. 7, Chap. 3, pp. 135-224. Elsevier Science, Amsterdam (1995).
21. Gogate, V., Dechter, R.: A complete anytime algorithm for treewidth. In: Proceedings of the 20th Annual Conference on Uncertainty in Artificial Intelligence UAI-04. Arlington, Virginia, USA, 2004, pp. 201-208. AUAI Press, Berkeley (2004).
22. Goldberg, A. V., Karzanov, A. V.: Maximum skew-symmetric flows and matchings. Math. Program. 100, 537-568 (2004).
23. Guo, J.: A more effective linear kernelization for cluster editing. In: Chen, B., Paterson, M., Zhang, G. (eds.) Proceedings First International Symposium on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies ESCAPE 2007. Lecture Notes in Computer Science, vol. 4614, pp. 36-47. Springer, Berlin (2007).
24. Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., Wernicke, S.: Improved fixed-parameter algorithms for two feedback set problems. In: Proc. 9th Int. Workshop on Algorithms and Data Structures WADS 2004. Lecture Notes in Computer Science, vol. 3608, pp. 158-168. Springer, Berlin (2004).
25. Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38, 31-45 (2007).
26. Kanj, I. A., Pelsmajer, M. J., Schaefer, M.: Parameterized algorithms for feedback vertex set. In: Downey, R. G., Fellows, M. R. (eds.) Proceedings 1st International Workshop on Parameterized and Exact Computation, IWPEC 2004. Lecture Notes in Computer Science, vol. 3162, pp. 235-248. Springer, Berlin (2004).
27. Karp, R. M.: Reducibility among combinatorial problems. In: Miller, R. E., Thatcher, J. W. (eds.) Complexity of Computer Computations, pp. 85-104. Plenum Press, New York (1972).
28. Mehlhorn, K., Näher, S.: LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1995).
29. Nagamochi, H., Ibaraki, T.: A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica 7, 583-596 (1992).
30. Nemhauser, G. L., Trotter, L. E.: Vertex packing: structural properties and algorithms. Math. Program. 8, 232-248 (1975).
31. Niedermeier, R.: Invitation to fixed-parameter algorithms. Universität Tübingen. Habilitation Thesis (2002).
32. Pearl, J.: Probablistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, Palo Alto (1988).
33. Raman, V., Saurabh, S., Subramanian, C. R.: Faster fixed parameter tractable algorithms for undirected feedback vertex set. In: Proceedings 13th International Symposium on Algorithms and Computation, ISAAC 2002. Lecture Notes in Computer Science, vol. 2518, pp. 241-248. Springer, Berlin (2002).
34. Raman, V., Saurabh, S., Subramanian, C. R.: Faster algorithms for feedback vertex set. Electronic Notes in Discrete Mathematics 19, 273-279 (2005). Proceedings 2nd Brazilian Symposium on Graphs, Algorithms, and Combinatorics, GRACO 2005.
35. Razgon, I.: Exact computation of maximum induced forest. In: Arge, L., Freivalds, R. (eds.) Proceedings of the 10th Scandinavian Workshop on Algorithm Theory, SWAT 2006. Lecture Notes in Computer Science, vol. 4059, pp. 160-171. Springer, Berlin (2006).
36. Schrijver, A.: Combinatorial Optimization. Polyhedra and Efficiency. Springer, Berlin (2003).
37. Thomassé, S.: A quadratic kernel for feedback vertex set. In: SODA '09: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Philadelphia, PA, USA, 2009. pp. 115-119. Society for Industrial and Applied Mathematics, Philadelphia (2009).
38. Treewidthlib. http://www.cs.uu.nl/people/hansb/treewidthlib (2004).
39. Tutte, W. T.: A short proof of the factor theorem for finite graphs. Can. J. Math. 6, 347-352 (1954).
40. van Dijk, T. C.: Fixed parameter complexity of feedback problems. Master's thesis, Utrecht University (2007).