This document discusses the kernelization complexity of finding colorful motifs in graphs. It introduces colorful motifs as a problem with applications in bioinformatics. It shows the problem is NP-complete even on very simple graph classes from the kernelization perspective. It presents an observation that leads to many polynomial kernels in a special case of comb graphs, but notes this does not generalize. It also provides NP-hardness results and observations ruling out polynomial kernels for more general graph classes and problems.

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 simple
graph classes.

5. An observation leading to many polynomial kernels in a very special
situation.

6. An observation leading to many polynomial kernels in a very special
situation.
More observations ruling out approaches towards many poly kernels in
more general situations.

7. An observation leading to many polynomial kernels in a very special
situation.
More observations ruling out approaches towards many poly kernels in
more general situations.
Some NP-hardness results with applications to discovering other
hardness results.

8. An observation leading to many polynomial kernels in a very special
situation.
More observations ruling out approaches towards many poly kernels in
more general situations.
Some NP-hardness results with applications to discovering other
hardness results.
Observing hardness of polynomial kernelization on other classes of
graphs.

9. Introducing Colorful Motifs
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10. Introducing Colorful Motifs
First: G M
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11. S
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12. S
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13. S
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14. Topology-free querying of protein interaction networks.
Sharon Bruckner, Falk Hüffner, Richard M. Karp, Ron Shamir, and
Roded Sharan.
In Proceedings of RECOMB .
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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
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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 
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17. NP-Complete even when:
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18. NP-Complete even when:
G is a tree with maximum degree .
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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 over
just two colors.
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20. FPT parameterized by |M|,
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21. FPT parameterized by |M|,
W[]-hard parameterized by the number of colors in M.
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22. C M
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23. S
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24. S
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25. S
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26. Finding and Counting Vertex-Colored Subtrees.
Sylvain Guillemot and Florian Sikora.
In MFCS .
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27. Finding and Counting Vertex-Colored Subtrees.
Sylvain Guillemot and Florian Sikora.
In MFCS .
A O∗ (2|M| ) algorithm.
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28. Kernelization hardness of connectivity problems in d-degenerate graphs.
Marek Cygan, Marcin Pilipczuk, Micha Pilipczuk, and Jakub O.
Wojtaszczyk.
In WG .
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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....
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30. S
comb graphs O
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31. No polynomial kernels¹.
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32. S
comb graphs - composition O
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33. S
comb graphs - composition O
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34. Many polynomial kernels.
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35. S
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36. S
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37. S
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38. S
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39. S
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40. S
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41. On comb graphs:
We get n kernels of size O(k2 ) each.
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42. On comb graphs:
We get n kernels of size O(k2 ) each.
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43. On what other classes can we obtain many polynomial kernels using this
observation?
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44. On what other classes can we obtain many polynomial kernels using this
observation?
Unfortunately, this observation does not take us very far.
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45. R C M does not admit a polynomial kernel on trees.
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46. “Fixing” a single vertex does not help the cause of kernelization.
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47. “Fixing” a single vertex does not help the cause of kernelization.
S C M does not admit a polynomial kernel on trees.
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48. “Fixing” a single vertex does not help the cause of kernelization.
“Fixing” a constant subset of vertices does not help either.
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49. Many polynomial kernels for trees? On more general classes of graphs?
E M
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N
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50. A framework for ruling out the possibility of many polynomial kernels?
E M
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51. Paths: Polynomial time.
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52. Catterpillars: Polynomial time.
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53. Lobsters: NP-hard even when...
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54. ...restricted to “superstar graphs”.
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55. “Hardness” on superstars → “Hardness” on graphs of diameter two.
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56. “Hardness” on superstars → “Hardness” on graphs of diameter two.
“Hardness” on general graphs → “Hardness” on graphs of diameter
three.
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57. “Hardness” on superstars → “Hardness” on graphs of diameter two.
“Hardness” on general graphs → “Hardness” on graphs of diameter
three.
NP-hardness
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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
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59. Polynomial kernels for graphs of diameter two? For superstars?
E M
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N
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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
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70. Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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71. Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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72. Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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73. Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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74. Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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75. Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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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 to
reduce 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 opportunity
for creativity: finding parameters that are sensible in practice and useful
for algorithms.
Concluding Remarks

80. Thank you!