kernelization complexity of colorful motifs
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TRANSCRIPT
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On the kernelization complexity
of colorful motifs
IPEC 2010
joint work withAbhimanyu M. Ambalath , Radheshyam Balasundaram ,
Chintan Rao H, Venkata Koppula, Geevarghese Philip , and M S Ramanujan
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Colorful Motifs.
A problem with immense utility in bioinformatics.Intractable — from the kernelization point of view — on very simplegraph classes.
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Colorful Motifs.
A problem with immense utility in bioinformatics.
Intractable — from the kernelization point of view — on very simplegraph classes.
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Colorful Motifs.
A problem with immense utility in bioinformatics.Intractable — from the kernelization point of view — on very simplegraph classes.
![Page 5: Kernelization Complexity of Colorful Motifs](https://reader033.vdocument.in/reader033/viewer/2022050919/547c698fb37959492b8b4fe9/html5/thumbnails/5.jpg)
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.
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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.
![Page 7: Kernelization Complexity of Colorful Motifs](https://reader033.vdocument.in/reader033/viewer/2022050919/547c698fb37959492b8b4fe9/html5/thumbnails/7.jpg)
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.
![Page 8: Kernelization Complexity of Colorful Motifs](https://reader033.vdocument.in/reader033/viewer/2022050919/547c698fb37959492b8b4fe9/html5/thumbnails/8.jpg)
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.
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Introducing Colorful Motifs
First: G M
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Introducing Colorful Motifs
First: G M
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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 andBioinformatics, .
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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 andBioinformatics, .
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Sharp tractability borderlines for finding connected motifs invertex-colored graphs.
Michael R. Fellows, Guillaume Fertin, Danny Hermelin, and StéphaneVialette.
In Proceedings of ICALP
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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.
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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.
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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.
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FPT parameterized by |M|,
W[]-hard parameterized by the number of colors in M.
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FPT parameterized by |M|,
W[]-hard parameterized by the number of colors in M.
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C M
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Finding and Counting Vertex-Colored Subtrees.Sylvain Guillemot and Florian Sikora.
In MFCS .
A O∗(2|M|) algorithm.
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Finding and Counting Vertex-Colored Subtrees.Sylvain Guillemot and Florian Sikora.
In MFCS .
A O∗(2|M|) algorithm.
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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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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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Scomb graphs
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No polynomial kernels¹.
¹Unless CoNP ⊆ NP/poly
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Scomb graphs - composition
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Scomb graphs - composition
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Many polynomial kernels.
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On comb graphs:We get n kernels of size O(k2) each.
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On comb graphs:We get n kernels of size O(k2) each.
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On what other classes can we obtain many polynomial kernels using thisobservation?
Unfortunately, this observation does not take us very far.
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On what other classes can we obtain many polynomial kernels using thisobservation?
Unfortunately, this observation does not take us very far.
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R C M does not admit a polynomial kernel on trees.
S C M does not admit a polynomial kernel on trees.“Fixing” a constant subset of vertices does not help either.
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“Fixing” a single vertex does not help the cause of kernelization.
S C M does not admit a polynomial kernel on trees.“Fixing” a constant subset of vertices does not help either.
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“Fixing” a single vertex does not help the cause of kernelization.
S C M does not admit a polynomial kernel on trees.
“Fixing” a constant subset of vertices does not help either.
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“Fixing” a single vertex does not help the cause of kernelization.
S C M does not admit a polynomial kernel on trees.
“Fixing” a constant subset of vertices does not help either.
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OPEN
PRO
BLEM
Many polynomial kernels for trees? On more general classes of graphs?
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OPEN
PRO
BLEM
A framework for ruling out the possibility of many polynomial kernels?
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Paths: Polynomial time.
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Catterpillars: Polynomial time.
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Lobsters: NP-hard even when...
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...restricted to “superstar graphs”.
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“Hardness” on superstars → “Hardness” on graphs of diameter two.
“Hardness” on general graphs → “Hardness” on graphs of diameterthree.
NP-hardness
NP-hardness and infeasibility of obtaining polynomial kernels
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“Hardness” on superstars → “Hardness” on graphs of diameter two.
“Hardness” on general graphs → “Hardness” on graphs of diameterthree.
NP-hardness
NP-hardness and infeasibility of obtaining polynomial kernels
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“Hardness” on superstars → “Hardness” on graphs of diameter two.
“Hardness” on general graphs → “Hardness” on graphs of diameterthree.
NP-hardness
NP-hardness and infeasibility of obtaining polynomial kernels
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COLO
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“Hardness” on superstars → “Hardness” on graphs of diameter two.
“Hardness” on general graphs → “Hardness” on graphs of diameterthree.
NP-hardness
NP-hardness and infeasibility of obtaining polynomial kernels
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OPEN
PRO
BLEM
Polynomial kernels for graphs of diameter two? For superstars?
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AN A
PPLICATIO
N
C D S G D O
Constant Time
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AN A
PPLICATIO
N
C D S G D O
Constant Time
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AN A
PPLICATIO
N
C D S G D T
Constant Time
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AN A
PPLICATIO
N
C D S G D T
W[]-hard (Split Graphs)
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AN A
PPLICATIO
N
C D S G D T
Constant Time
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AN A
PPLICATIO
N
C D S G D T
???
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AN A
PPLICATIO
N
C D S G D T
NP-complete
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AN A
PPLICATIO
N
C D S G D T
W[]-hard
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AN A
PPLICATIO
N
Colorful Motifs on graphs of diameter two↓Connected Dominating Set on graphs of diameter two
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Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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Reduction from Colorful Set Cover
to Colorful Motifs on Superstars
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Concluding Remarks
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 evenNP-hardness, for graph problems that have a connectivity requirement
from the solution.
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Concluding Remarks
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 evenNP-hardness, for graph problems that have a connectivity requirement
from the solution.
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Concluding Remarks
Several open questions in the context of kernelization for the colorfulmotifs problem alone, and also for closely related problems.
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Concluding Remarks
When we encounter hardness, we are compelled to look out for otherparameters for improving the situation. ere is plenty of opportunityfor creativity: finding parameters that are sensible in practice and useful
for algorithms.
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Thank you!