Kernelization Complexity of Colorful Motifs

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Kernelization Complexity of Colorful Motifs

  1. 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. 2. Colorful Motifs.
  3. 3. Colorful Motifs.A problem with immense utility in bioinformatics.
  4. 4. Colorful Motifs.A problem with immense utility in bioinformatics.Intractable — from the kernelization point of view — on very simplegraph classes.
  5. 5. An observation leading to many polynomial kernels in a very specialsituation.
  6. 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. 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. 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. 9. Introducing Colorful Motifs S TIF O M L RFU LO CO
  10. 10. Introducing Colorful Motifs First: G M S TIF O M L RFU LO CO
  11. 11. S TIF O M L RFU LOCO
  12. 12. S TIF O M L RFU LOCO
  13. 13. S TIF O M L RFU LOCO
  14. 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. 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. 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. 17. NP-Complete even when: S TIF O M L RFU LO CO
  18. 18. NP-Complete even when:G is a tree with maximum degree . S TIF O M L RFU LO CO
  19. 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. 20. FPT parameterized by |M|, S TIF O M L RFU LO CO
  21. 21. FPT parameterized by |M|,W[]-hard parameterized by the number of colors in M. S TIF O M L RFU LO CO
  22. 22. C M S TIF O M L RFU LO CO
  23. 23. S TIF O M L RFU LOCO
  24. 24. S TIF O M L RFU LOCO
  25. 25. S TIF O M L RFU LOCO
  26. 26. Finding and Counting Vertex-Colored Subtrees. Sylvain Guillemot and Florian Sikora. In MFCS . S TIF O M L RFU LO CO
  27. 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. 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. 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. 30. Scomb graphs O TIF M L RFU LO CO
  31. 31. No polynomial kernels¹. S TIF O M L RFU LO¹Unless CoNP ⊆ NP/poly CO
  32. 32. Scomb graphs - composition O TIF M L RFU LO CO
  33. 33. Scomb graphs - composition O TIF M L RFU LO CO
  34. 34. Many polynomial kernels. S TIF O M L RFU LO CO
  35. 35. S TIF O M L RFU LOCO
  36. 36. S TIF O M L RFU LOCO
  37. 37. S TIF O M L RFU LOCO
  38. 38. S TIF O M L RFU LOCO
  39. 39. S TIF O M L RFU LOCO
  40. 40. S TIF O M L RFU LOCO
  41. 41. On comb graphs:We get n kernels of size O(k2 ) each. S TIF O M L RFU LO CO
  42. 42. On comb graphs:We get n kernels of size O(k2 ) each. S TIF O M L RFU LO CO
  43. 43. On what other classes can we obtain many polynomial kernels using this observation? S TIF O M L RFU LO CO
  44. 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. 45. R C M does not admit a polynomial kernel on trees. S TIF O M L RFU LO CO
  46. 46. “Fixing” a single vertex does not help the cause of kernelization. S TIF O M L RFU LO CO
  47. 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. 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. 49. Many polynomial kernels for trees? On more general classes of graphs? E M BL O PR N PE O
  50. 50. A framework for ruling out the possibility of many polynomial kernels? E M BL O PR N PE O
  51. 51. Paths: Polynomial time. S TIF O M L RFU LO CO
  52. 52. Catterpillars: Polynomial time. S TIF O M L RFU LO CO
  53. 53. Lobsters: NP-hard even when... S TIF O M L RFU LO CO
  54. 54. ...restricted to “superstar graphs”. S TIF O M L RFU LO CO
  55. 55. “Hardness” on superstars → “Hardness” on graphs of diameter two. S TIF O M L RFU LO CO
  56. 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. 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. 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. 59. Polynomial kernels for graphs of diameter two? For superstars? E M BL O PR N PE O
  60. 60. C D S G  D O Constant Time N IO AT L IC APP AN
  61. 61. C D S G  D O Constant Time N IO AT L IC APP AN
  62. 62. C D S G  D T Constant Time N IO AT L IC APP AN
  63. 63. C D S G  D T W[]-hard (Split Graphs) N IO AT L IC APP AN
  64. 64. C D S G  D T Constant Time N IO AT L IC APP AN
  65. 65. C D S G  D T ??? N IO AT L IC APP AN
  66. 66. C D S G  D T NP-complete N IO AT L IC APP AN
  67. 67. C D S G  D T W[]-hard N IO AT L IC APP AN
  68. 68. Colorful Motifs on graphs of diameter two ↓Connected Dominating Set on graphs of diameter two N IO AT L IC APP AN
  69. 69. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
  70. 70. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
  71. 71. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
  72. 72. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
  73. 73. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
  74. 74. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
  75. 75. Reduction from Colorful Set Cover to Colorful Motifs on Superstars S TIF O M L RFU LO CO
  76. 76. is was an advertisement for C M.Concluding Remarks
  77. 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. 78. Several open questions in the context of kernelization for the colorful motifs problem alone, and also for closely related problems. Concluding Remarks
  79. 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. 80. Thank you!

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