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- 1. On the kernelization complexity of colorful motifs IPEC 2010 joint work with Abhimanyu M. Ambalath , Radheshyam Balasundaram ,Chintan Rao H, Venkata Koppula, Geevarghese Philip , and M S Ramanujan
- 2. Colorful Motifs.
- 3. Colorful Motifs.A problem with immense utility in bioinformatics.
- 4. Colorful Motifs.A problem with immense utility in bioinformatics.Intractable — from the kernelization point of view — on very simplegraph classes.
- 5. An observation leading to many polynomial kernels in a very specialsituation.
- 6. An observation leading to many polynomial kernels in a very specialsituation.More observations ruling out approaches towards many poly kernels inmore general situations.
- 7. An observation leading to many polynomial kernels in a very specialsituation.More observations ruling out approaches towards many poly kernels inmore general situations.Some NP-hardness results with applications to discovering otherhardness results.
- 8. An observation leading to many polynomial kernels in a very specialsituation.More observations ruling out approaches towards many poly kernels inmore general situations.Some NP-hardness results with applications to discovering otherhardness results.Observing hardness of polynomial kernelization on other classes ofgraphs.
- 9. Introducing Colorful Motifs S TIF O M L RFU LO CO
- 10. Introducing Colorful Motifs First: G M S TIF O M L RFU LO CO
- 11. S TIF O M L RFU LOCO
- 12. S TIF O M L RFU LOCO
- 13. S TIF O M L RFU LOCO
- 14. Topology-free querying of protein interaction networks.Sharon Bruckner, Falk Hüﬀner, Richard M. Karp, Ron Shamir, and Roded Sharan. In Proceedings of RECOMB . S TIF O M L RFU LO CO
- 15. Topology-free querying of protein interaction networks.Sharon Bruckner, Falk Hüﬀner, Richard M. Karp, Ron Shamir, and Roded Sharan. In Proceedings of RECOMB . Motif search in graphs: Application to metabolic networks. Vincent Lacroix, Cristina G. Fernandes, and Marie-France Sagot. IEEE/ACM Transactions on Computational Biology and Bioinformatics, . S TIF O M L RFU LO CO
- 16. Sharp tractability borderlines for ﬁnding connected motifs in vertex-colored graphs.Michael R. Fellows, Guillaume Fertin, Danny Hermelin, and Stéphane Vialette. In Proceedings of ICALP S TIF O M L RFU LO CO
- 17. NP-Complete even when: S TIF O M L RFU LO CO
- 18. NP-Complete even when:G is a tree with maximum degree . S TIF O M L RFU LO CO
- 19. NP-Complete even when:G is a tree with maximum degree .G is a bipartite graph with maximum degree and M is a multiset overjust two colors. S TIF O M L RFU LO CO
- 20. FPT parameterized by |M|, S TIF O M L RFU LO CO
- 21. FPT parameterized by |M|,W[]-hard parameterized by the number of colors in M. S TIF O M L RFU LO CO
- 22. C M S TIF O M L RFU LO CO
- 23. S TIF O M L RFU LOCO
- 24. S TIF O M L RFU LOCO
- 25. S TIF O M L RFU LOCO
- 26. Finding and Counting Vertex-Colored Subtrees. Sylvain Guillemot and Florian Sikora. In MFCS . S TIF O M L RFU LO CO
- 27. Finding and Counting Vertex-Colored Subtrees. Sylvain Guillemot and Florian Sikora. In MFCS . A O∗ (2|M| ) algorithm. S TIF O M L RFU LO CO
- 28. Kernelization hardness of connectivity problems in d-degenerate graphs. Marek Cygan, Marcin Pilipczuk, Micha Pilipczuk, and Jakub O. Wojtaszczyk. In WG . S TIF O M L RFU LO CO
- 29. Kernelization hardness of connectivity problems in d-degenerate graphs. Marek Cygan, Marcin Pilipczuk, Micha Pilipczuk, and Jakub O. Wojtaszczyk. In WG . NP-complete even when restricted to.... S TIF O M L RFU LO CO
- 30. Scomb graphs O TIF M L RFU LO CO
- 31. No polynomial kernels¹. S TIF O M L RFU LO¹Unless CoNP ⊆ NP/poly CO
- 32. Scomb graphs - composition O TIF M L RFU LO CO
- 33. Scomb graphs - composition O TIF M L RFU LO CO
- 34. Many polynomial kernels. S TIF O M L RFU LO CO
- 35. S TIF O M L RFU LOCO
- 36. S TIF O M L RFU LOCO
- 37. S TIF O M L RFU LOCO
- 38. S TIF O M L RFU LOCO
- 39. S TIF O M L RFU LOCO
- 40. S TIF O M L RFU LOCO
- 41. On comb graphs:We get n kernels of size O(k2 ) each. S TIF O M L RFU LO CO
- 42. On comb graphs:We get n kernels of size O(k2 ) each. S TIF O M L RFU LO CO
- 43. On what other classes can we obtain many polynomial kernels using this observation? S TIF O M L RFU LO CO
- 44. On what other classes can we obtain many polynomial kernels using this observation? Unfortunately, this observation does not take us very far. S TIF O M L RFU LO CO
- 45. R C M does not admit a polynomial kernel on trees. S TIF O M L RFU LO CO
- 46. “Fixing” a single vertex does not help the cause of kernelization. S TIF O M L RFU LO CO
- 47. “Fixing” a single vertex does not help the cause of kernelization.S C M does not admit a polynomial kernel on trees. S TIF O M L RFU LO CO
- 48. “Fixing” a single vertex does not help the cause of kernelization. “Fixing” a constant subset of vertices does not help either. S TIF O M L RFU LO CO
- 49. Many polynomial kernels for trees? On more general classes of graphs? E M BL O PR N PE O
- 50. A framework for ruling out the possibility of many polynomial kernels? E M BL O PR N PE O
- 51. Paths: Polynomial time. S TIF O M L RFU LO CO
- 52. Catterpillars: Polynomial time. S TIF O M L RFU LO CO
- 53. Lobsters: NP-hard even when... S TIF O M L RFU LO CO
- 54. ...restricted to “superstar graphs”. S TIF O M L RFU LO CO
- 55. “Hardness” on superstars → “Hardness” on graphs of diameter two. S TIF O M L RFU LO CO
- 56. “Hardness” on superstars → “Hardness” on graphs of diameter two.“Hardness” on general graphs → “Hardness” on graphs of diameter three. S TIF O M L RFU LO CO
- 57. “Hardness” on superstars → “Hardness” on graphs of diameter two.“Hardness” on general graphs → “Hardness” on graphs of diameter three. NP-hardness S TIF O M L RFU LO CO
- 58. “Hardness” on superstars → “Hardness” on graphs of diameter two.“Hardness” on general graphs → “Hardness” on graphs of diameter three. NP-hardness NP-hardness and infeasibility of obtaining polynomial kernels S TIF O M L RFU LO CO
- 59. Polynomial kernels for graphs of diameter two? For superstars? E M BL O PR N PE O
- 60. C D S G D O Constant Time N IO AT L IC APP AN
- 61. C D S G D O Constant Time N IO AT L IC APP AN
- 62. C D S G D T Constant Time N IO AT L IC APP AN
- 63. C D S G D T W[]-hard (Split Graphs) N IO AT L IC APP AN
- 64. C D S G D T Constant Time N IO AT L IC APP AN
- 65. C D S G D T ??? N IO AT L IC APP AN
- 66. C D S G D T NP-complete N IO AT L IC APP AN
- 67. C D S G D T W[]-hard N IO AT L IC APP AN
- 68. Colorful Motifs on graphs of diameter two ↓Connected Dominating Set on graphs of diameter two N IO AT L IC APP AN
- 69. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
- 70. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
- 71. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
- 72. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
- 73. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
- 74. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
- 75. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
- 76. is was an advertisement for C M.Concluding Remarks
- 77. is was an advertisement for C M. We wish to propose that it may be a popular choice as a problem toreduce from, for showing hardness of polynomial kernelization, and even NP-hardness, for graph problems that have a connectivity requirement from the solution. Concluding Remarks
- 78. Several open questions in the context of kernelization for the colorful motifs problem alone, and also for closely related problems. Concluding Remarks
- 79. When we encounter hardness, we are compelled to look out for other parameters for improving the situation. ere is plenty of opportunityfor creativity: ﬁnding parameters that are sensible in practice and useful for algorithms. Concluding Remarks
- 80. Thank you!

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