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

Colorful Motifs.

A problem with immense utility in bioinformatics.Intractable — from the kernelization point of view — on very simplegraph classes.

Colorful Motifs.

A problem with immense utility in bioinformatics.

Intractable — from the kernelization point of view — on very simplegraph classes.

Colorful Motifs.

A problem with immense utility in bioinformatics.Intractable — from the kernelization point of view — on very simplegraph classes.

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.

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.

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.

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.

OPEN

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Many polynomial kernels for trees? On more general classes of graphs?

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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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“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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Polynomial kernels for graphs of diameter two? For superstars?

AN A

PPLICATIO

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C D S G D O

Constant Time

AN A

PPLICATIO

N

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Constant Time

AN A

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Constant Time

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C D S G D T

W[]-hard (Split Graphs)

AN A

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C D S G D T

Constant Time

AN A

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C D S G D T

???

AN A

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NP-complete

AN A

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W[]-hard

AN A

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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

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.

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.

Concluding Remarks

Several open questions in the context of kernelization for the colorfulmotifs problem alone, and also for closely related problems.

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.

Thank you!