[PPT] - Kernel lower bounds using co-nondeterminism: Finding induced PowerPoint Presentation

SLIDE 1

Kernel lower bounds using co-nondeterminism: Finding induced hereditary subgraphs

Stefan Kratsch, Marcin Pilipczuk, Ashutosh Rai, Venkatesh Raman SWAT 2012, Helsinki

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 1/12

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Kernelization

instance of NP-hard problem

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 2/12

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Kernelization

instance of NP-hard problem k

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 2/12

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Kernelization

k instance of NP-hard problem

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 2/12

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Kernelization

k instance of NP-hard problem

poly time

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 2/12

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Kernelization

k instance of NP-hard problem

poly time

size ≤ g(k)

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 2/12

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Kernelization

k instance of NP-hard problem

poly time

size ≤ g(k) small: g(k) polynomial

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 2/12

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Kernelization

k instance of NP-hard problem

poly time

size ≤ g(k) small: g(k) polynomial sometimes impossible

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 2/12

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Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08]

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

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Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances

f hard language L
S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

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Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances

f hard language L

poly time

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

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Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances

f hard language L

poly time

k

OR instance

f param. lang. Q
S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

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Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances

f hard language L

poly time

k

OR instance

f param. lang. Q

poly time

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

SLIDE 14

Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances

f hard language L

poly time

k

OR instance

f param. lang. Q

poly time size ≤ poly(k)

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

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Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances

f hard language L

poly time

k

OR instance

f param. lang. Q

poly time size ≤ poly(k) [Fortnow, Santhanam, STOC’08] ⇒ L ∈ coNP/poly

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

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Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances

f hard language L

poly time

k

OR instance

f param. lang. Q

poly time size ≤ poly(k) [Fortnow, Santhanam, STOC’08] ⇒ L ∈ coNP/poly L is NP-hard ⇒ NP ⊆ coNP/poly ⇒ PH = ΣP

3

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

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Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances

f hard language L

coNP time

k

OR instance

f param. lang. Q

coNP time size ≤ poly(k) [Fortnow, Santhanam, STOC’08] ⇒ L ∈ coNP/poly L is NP-hard ⇒ NP ⊆ coNP/poly ⇒ PH = ΣP

3

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

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Kernelization lower bounds

[Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances

f hard language L

coNP time

k

OR instance

f param. lang. Q

coNP time size ≤ poly(k) [Fortnow, Santhanam, STOC’08] ⇒ L ∈ coNP/poly L is NP-hard ⇒ NP ⊆ coNP/poly ⇒ PH = ΣP

3

under “coNP reductions”

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 3/12

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Kernelization lower bounds with co-nondeterminism

Start with a language L, that is “NP-hard under co-nondeterministic many-one reductions”.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 4/12

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Kernelization lower bounds with co-nondeterminism

Start with a language L, that is “NP-hard under co-nondeterministic many-one reductions”.

I.e., ∃ nondeterministic poly-time reduction from NP-hard ¯ L to L, that

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 4/12

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Kernelization lower bounds with co-nondeterminism

Start with a language L, that is “NP-hard under co-nondeterministic many-one reductions”.

I.e., ∃ nondeterministic poly-time reduction from NP-hard ¯ L to L, that if input ∈ ¯ L, then all outputs ∈ L, and

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 4/12

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Kernelization lower bounds with co-nondeterminism

Start with a language L, that is “NP-hard under co-nondeterministic many-one reductions”.

I.e., ∃ nondeterministic poly-time reduction from NP-hard ¯ L to L, that if input ∈ ¯ L, then all outputs ∈ L, and if input / ∈ ¯ L, then on at least one computation path output / ∈ L.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 4/12

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Kernelization lower bounds with co-nondeterminism

Start with a language L, that is “NP-hard under co-nondeterministic many-one reductions”.

I.e., ∃ nondeterministic poly-time reduction from NP-hard ¯ L to L, that if input ∈ ¯ L, then all outputs ∈ L, and if input / ∈ ¯ L, then on at least one computation path output / ∈ L.

Compose t instances xi in co-nondeterministic poly-time into one OR-instance of Q with parameter ≤ poly(maxi |xi|)to(1).

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 4/12

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Kernelization lower bounds with co-nondeterminism

Start with a language L, that is “NP-hard under co-nondeterministic many-one reductions”.

I.e., ∃ nondeterministic poly-time reduction from NP-hard ¯ L to L, that if input ∈ ¯ L, then all outputs ∈ L, and if input / ∈ ¯ L, then on at least one computation path output / ∈ L.

Compose t instances xi in co-nondeterministic poly-time into one OR-instance of Q with parameter ≤ poly(maxi |xi|)to(1).

That is, if at least one xi ∈ L, then all outputs ∈ Q.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 4/12

SLIDE 25

Kernelization lower bounds with co-nondeterminism

Start with a language L, that is “NP-hard under co-nondeterministic many-one reductions”.

I.e., ∃ nondeterministic poly-time reduction from NP-hard ¯ L to L, that if input ∈ ¯ L, then all outputs ∈ L, and if input / ∈ ¯ L, then on at least one computation path output / ∈ L.

Compose t instances xi in co-nondeterministic poly-time into one OR-instance of Q with parameter ≤ poly(maxi |xi|)to(1).

That is, if at least one xi ∈ L, then all outputs ∈ Q. If all xi / ∈ L, then on at least one computation path output / ∈ Q.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 4/12

SLIDE 26

Kernelization lower bounds with co-nondeterminism

Start with a language L, that is “NP-hard under co-nondeterministic many-one reductions”.

I.e., ∃ nondeterministic poly-time reduction from NP-hard ¯ L to L, that if input ∈ ¯ L, then all outputs ∈ L, and if input / ∈ ¯ L, then on at least one computation path output / ∈ L.

Compose t instances xi in co-nondeterministic poly-time into one OR-instance of Q with parameter ≤ poly(maxi |xi|)to(1).

That is, if at least one xi ∈ L, then all outputs ∈ Q. If all xi / ∈ L, then on at least one computation path output / ∈ Q.

Then, a (co-nondeterministic) polynomial kernelization of Q implies NP ⊆ coNP/poly.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 4/12

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Case study: Π-Induced Subgraph

Co-nondeterminism first used by Kratsch (SODA’12) for Ramsey.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 5/12

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Case study: Π-Induced Subgraph

Co-nondeterminism first used by Kratsch (SODA’12) for Ramsey.

Does a graph G has a clique or independent size of size k?

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 5/12

SLIDE 29

Case study: Π-Induced Subgraph

Co-nondeterminism first used by Kratsch (SODA’12) for Ramsey.

Does a graph G has a clique or independent size of size k?

Our work: generalize to most important cases of Π-Induced Subgraph.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 5/12

SLIDE 30

Case study: Π-Induced Subgraph

Co-nondeterminism first used by Kratsch (SODA’12) for Ramsey.

Does a graph G has a clique or independent size of size k?

Our work: generalize to most important cases of Π-Induced Subgraph. Π-Induced Subgraph

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 5/12

SLIDE 31

Case study: Π-Induced Subgraph

Co-nondeterminism first used by Kratsch (SODA’12) for Ramsey.

Does a graph G has a clique or independent size of size k?

Our work: generalize to most important cases of Π-Induced Subgraph. Π-Induced Subgraph Input: A graph G and a parameter k.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 5/12

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Case study: Π-Induced Subgraph

Co-nondeterminism first used by Kratsch (SODA’12) for Ramsey.

Does a graph G has a clique or independent size of size k?

Our work: generalize to most important cases of Π-Induced Subgraph. Π-Induced Subgraph Input: A graph G and a parameter k. Question: Does there exist an induced subgraph of G on k vertices that belongs to Π?

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 5/12

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Case study: Π-Induced Subgraph

Co-nondeterminism first used by Kratsch (SODA’12) for Ramsey.

Does a graph G has a clique or independent size of size k?

Our work: generalize to most important cases of Π-Induced Subgraph. Π-Induced Subgraph Input: A graph G and a parameter k. Question: Does there exist an induced subgraph of G on k vertices that belongs to Π? [Khot, Raman, 2000] ⇒ Π-Induced Subgraph is FPT iff Π contains all cliques and independent sets.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 5/12

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Case study: Π-Induced Subgraph

Co-nondeterminism first used by Kratsch (SODA’12) for Ramsey.

Does a graph G has a clique or independent size of size k?

Our work: generalize to most important cases of Π-Induced Subgraph. Π-Induced Subgraph Input: A graph G and a parameter k. Question: Does there exist an induced subgraph of G on k vertices that belongs to Π? [Khot, Raman, 2000] ⇒ Π-Induced Subgraph is FPT iff Π contains all cliques and independent sets. Note: Ramsey = {cliques, ind. sets}-Induced Subgraph.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 5/12

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Π-Induced Subgraph

co-nondeterministic NP-hardness

Π-Induced Subgraph NP-hard unless trivial [Lewis, Yannakakis, 1980].

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 6/12

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Π-Induced Subgraph

co-nondeterministic NP-hardness

Π-Induced Subgraph NP-hard unless trivial [Lewis, Yannakakis, 1980]. For composition: much easier with improvement version. Improvement Π-Induced Subgraph

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 6/12

SLIDE 37

Π-Induced Subgraph

co-nondeterministic NP-hardness

Π-Induced Subgraph NP-hard unless trivial [Lewis, Yannakakis, 1980]. For composition: much easier with improvement version. Improvement Π-Induced Subgraph Input: A graph G, a parameter k, and a set X ⊆ V (G) of size k − 1 such that G[X] ∈ Π.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 6/12

SLIDE 38

Π-Induced Subgraph

co-nondeterministic NP-hardness

Π-Induced Subgraph NP-hard unless trivial [Lewis, Yannakakis, 1980]. For composition: much easier with improvement version. Improvement Π-Induced Subgraph Input: A graph G, a parameter k, and a set X ⊆ V (G) of size k − 1 such that G[X] ∈ Π. Question: Does there exist an induced subgraph of G on k vertices that belongs to Π?

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 6/12

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Π-Induced Subgraph

co-nondeterministic NP-hardness

Π-Induced Subgraph NP-hard unless trivial [Lewis, Yannakakis, 1980]. For composition: much easier with improvement version. Improvement Π-Induced Subgraph Input: A graph G, a parameter k, and a set X ⊆ V (G) of size k − 1 such that G[X] ∈ Π. Question: Does there exist an induced subgraph of G on k vertices that belongs to Π? Not so easy to get Karp-style NP-hardness for Improvement Π-Induced Subgraph.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 6/12

SLIDE 40

Π-Induced Subgraph

co-nondeterministic NP-hardness

Π-Induced Subgraph NP-hard unless trivial [Lewis, Yannakakis, 1980]. For composition: much easier with improvement version. Improvement Π-Induced Subgraph Input: A graph G, a parameter k, and a set X ⊆ V (G) of size k − 1 such that G[X] ∈ Π. Question: Does there exist an induced subgraph of G on k vertices that belongs to Π? Not so easy to get Karp-style NP-hardness for Improvement Π-Induced Subgraph. But we need only co-nondeterministic reduction! That is trivial.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 6/12

SLIDE 41

Π-Induced Subgraph

co-nondeterministic NP-hardness

Π-Induced Subgraph NP-hard unless trivial [Lewis, Yannakakis, 1980]. For composition: much easier with improvement version. Improvement Π-Induced Subgraph Input: A graph G, a parameter k, and a set X ⊆ V (G) of size k − 1 such that G[X] ∈ Π. Question: Does there exist an induced subgraph of G on k vertices that belongs to Π? Not so easy to get Karp-style NP-hardness for Improvement Π-Induced Subgraph. But we need only co-nondeterministic reduction! That is trivial.

Guess minimum 1 ≤ k′ ≤ k such that (G, k′) is a Π-Induced Subgraph NO-instance and guess X of size k′ − 1 such that G[X] ∈ Π.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 6/12

SLIDE 42

Π-Induced Subgraph

co-nondeterministic NP-hardness

Π-Induced Subgraph NP-hard unless trivial [Lewis, Yannakakis, 1980]. For composition: much easier with improvement version. Improvement Π-Induced Subgraph Input: A graph G, a parameter k, and a set X ⊆ V (G) of size k − 1 such that G[X] ∈ Π. Question: Does there exist an induced subgraph of G on k vertices that belongs to Π? Not so easy to get Karp-style NP-hardness for Improvement Π-Induced Subgraph. But we need only co-nondeterministic reduction! That is trivial.

Guess minimum 1 ≤ k′ ≤ k such that (G, k′) is a Π-Induced Subgraph NO-instance and guess X of size k′ − 1 such that G[X] ∈ Π. Note: we need Π to be poly-recognizable.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 6/12

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Π-Induced Subgraph

co-nondeterministic compositions

Assume Π is closed under embedding.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 7/12

SLIDE 44

Π-Induced Subgraph

co-nondeterministic compositions

Assume Π is closed under embedding.

G H

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 7/12

SLIDE 45

Π-Induced Subgraph

co-nondeterministic compositions

Assume Π is closed under embedding.

G H v

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 7/12

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Π-Induced Subgraph

co-nondeterministic compositions

Assume Π is closed under embedding.

Embed(G, v → H) H

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 7/12

SLIDE 47

Π-Induced Subgraph

co-nondeterministic compositions

Assume Π is closed under embedding.

Embed(G, v → H)

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 7/12

SLIDE 48

Π-Induced Subgraph

co-nondeterministic compositions

H1 H2 H3 H4 H5 t instances of Improvement-Π-IS equal parameter k

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 8/12

SLIDE 49

Π-Induced Subgraph

co-nondeterministic compositions

H1 H2 H3 H4 H5 t instances of Improvement-Π-IS equal parameter k host graph with t vertices

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 8/12

SLIDE 50

Π-Induced Subgraph

co-nondeterministic compositions

H1 H2 H3 H4 H5 t instances of Improvement-Π-IS equal parameter k host graph with t vertices each vertex ∈ ℓ-vertex Π-ind. subgr.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 8/12

SLIDE 51

Π-Induced Subgraph

co-nondeterministic compositions

H1 H2 H3 H4 H5 t instances of Improvement-Π-IS equal parameter k host graph with t vertices each vertex ∈ ℓ-vertex Π-ind. subgr. no (ℓ + 1)-vertex Π-ind. subgr.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 8/12

SLIDE 52

Π-Induced Subgraph

co-nondeterministic compositions

H1 H2 H3 H4 H5 t instances of Improvement-Π-IS equal parameter k host graph with t vertices each vertex ∈ ℓ-vertex Π-ind. subgr. no (ℓ + 1)-vertex Π-ind. subgr.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 8/12

SLIDE 53

Π-Induced Subgraph

co-nondeterministic compositions

H1 H2 H3 H4 H5 t instances of Improvement-Π-IS equal parameter k host graph with t vertices each vertex ∈ ℓ-vertex Π-ind. subgr. no (ℓ + 1)-vertex Π-ind. subgr. k′ = ℓ(k − 1) + 1

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 8/12

SLIDE 54

Π-Induced Subgraph

co-nondeterministic compositions

Need a host graph!

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 9/12

SLIDE 55

Π-Induced Subgraph

co-nondeterministic compositions

Need a host graph!

Crucial property: ℓ = to(1).

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 9/12

SLIDE 56

Π-Induced Subgraph

co-nondeterministic compositions

Need a host graph!

Crucial property: ℓ = to(1).

As in [Kratsch, SODA 2012]: use co-nondeterminism!

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 9/12

SLIDE 57

Π-Induced Subgraph

co-nondeterministic compositions

Need a host graph!

Crucial property: ℓ = to(1).

As in [Kratsch, SODA 2012]: use co-nondeterminism!

One catch: need proof of existence.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 9/12

SLIDE 58

Π-Induced Subgraph

co-nondeterministic compositions

Need a host graph!

Crucial property: ℓ = to(1).

As in [Kratsch, SODA 2012]: use co-nondeterminism!

One catch: need proof of existence.

Solution: Erd˝

s-Hajnal property.
S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 9/12

SLIDE 59

Π-Induced Subgraph

co-nondeterministic compositions

Need a host graph!

Crucial property: ℓ = to(1).

As in [Kratsch, SODA 2012]: use co-nondeterminism!

One catch: need proof of existence.

Solution: Erd˝

s-Hajnal property.

Erd˝

s-Hajnal conjecture

For any (not cofinite) hereditary graph property Π, there exists ε = ε(Π), such that any G ∈ Π has an independent set or a clique of size nε.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 9/12

SLIDE 60

Π-Induced Subgraph

co-nondeterministic compositions

Need a host graph!

Crucial property: ℓ = to(1).

As in [Kratsch, SODA 2012]: use co-nondeterminism!

One catch: need proof of existence.

Solution: Erd˝

s-Hajnal property.

Erd˝

s-Hajnal conjecture

For any (not cofinite) hereditary graph property Π, there exists ε = ε(Π), such that any G ∈ Π has an independent set or a clique of size nε. Open, proven for several special classes, such as perfect or Ks,s-free graphs.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 9/12

SLIDE 61

Π-Induced Subgraph

co-nondeterministic compositions

Need a host graph!

Crucial property: ℓ = to(1).

As in [Kratsch, SODA 2012]: use co-nondeterminism!

One catch: need proof of existence.

Solution: Erd˝

s-Hajnal property.

Erd˝

s-Hajnal conjecture

For any (not cofinite) hereditary graph property Π, there exists ε = ε(Π), such that any G ∈ Π has an independent set or a clique of size nε. Open, proven for several special classes, such as perfect or Ks,s-free graphs. Π has the Erd˝

s-Hajnal property ⇒ good host graph exists and we can find it

in coNP-time.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 9/12

SLIDE 62

Summary of results

Summarizing, we can prove that:

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 10/12

SLIDE 63

Summary of results

Summarizing, we can prove that: Theorem No poly-kernel for Π-Induced Subgraph for any non-trivial poly-recognizable hereditary graph class Π that contains all independent sets and cliques, is closed under embedding and has the Erd˝

s-Hajnal property.
S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 10/12

SLIDE 64

Summary of results

Summarizing, we can prove that: Theorem No poly-kernel for Π-Induced Subgraph for any non-trivial poly-recognizable hereditary graph class Π that contains all independent sets and cliques, is closed under embedding and has the Erd˝

s-Hajnal property.

Includes perfect graphs, cographs and permutation graphs.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 10/12

SLIDE 65

Summary of results

Summarizing, we can prove that: Theorem No poly-kernel for Π-Induced Subgraph for any non-trivial poly-recognizable hereditary graph class Π that contains all independent sets and cliques, is closed under embedding and has the Erd˝

s-Hajnal property.

Includes perfect graphs, cographs and permutation graphs. Using a few more tricks or by a reduction from Ramsey:

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 10/12

SLIDE 66

Summary of results

Summarizing, we can prove that: Theorem No poly-kernel for Π-Induced Subgraph for any non-trivial poly-recognizable hereditary graph class Π that contains all independent sets and cliques, is closed under embedding and has the Erd˝

s-Hajnal property.

Includes perfect graphs, cographs and permutation graphs. Using a few more tricks or by a reduction from Ramsey: Theorem No poly-kernel for Π-Induced Subgraph for any non-trivial poly-recognizable hereditary graph class Π that contains all independent sets, but excludes a certain biclique.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 10/12

SLIDE 67

Summary of results

Summarizing, we can prove that: Theorem No poly-kernel for Π-Induced Subgraph for any non-trivial poly-recognizable hereditary graph class Π that contains all independent sets and cliques, is closed under embedding and has the Erd˝

s-Hajnal property.

Includes perfect graphs, cographs and permutation graphs. Using a few more tricks or by a reduction from Ramsey: Theorem No poly-kernel for Π-Induced Subgraph for any non-trivial poly-recognizable hereditary graph class Π that contains all independent sets, but excludes a certain biclique. Includes chordal, interval, unit interval, claw-free and split graphs.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 10/12

SLIDE 68

Summary of results

Summarizing, we can prove that: Theorem No poly-kernel for Π-Induced Subgraph for any non-trivial poly-recognizable hereditary graph class Π that contains all independent sets and cliques, is closed under embedding and has the Erd˝

s-Hajnal property.

Includes perfect graphs, cographs and permutation graphs. Using a few more tricks or by a reduction from Ramsey: Theorem No poly-kernel for Π-Induced Subgraph for any non-trivial poly-recognizable hereditary graph class Π that contains all independent sets, but excludes a certain biclique. Includes chordal, interval, unit interval, claw-free and split graphs. Note: excluding a biclique implies the Erd˝

s-Hajnal property.
S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 10/12

SLIDE 69

Conclusions

We have shown kernelization hardness for most natural FPT-cases of Π-Induced Subgraph.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 11/12

SLIDE 70

Conclusions

We have shown kernelization hardness for most natural FPT-cases of Π-Induced Subgraph. Can we obtain a general lower bound for all FPT-cases of Π-Induced Subgraph?

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 11/12

SLIDE 71

Conclusions

We have shown kernelization hardness for most natural FPT-cases of Π-Induced Subgraph. Can we obtain a general lower bound for all FPT-cases of Π-Induced Subgraph? Interesting case: AT-free graphs.

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 11/12

SLIDE 72

Conclusions

We have shown kernelization hardness for most natural FPT-cases of Π-Induced Subgraph. Can we obtain a general lower bound for all FPT-cases of Π-Induced Subgraph? Interesting case: AT-free graphs.

Closed under embedding, so would fall under the first theorem. . .

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 11/12

SLIDE 73

Conclusions

We have shown kernelization hardness for most natural FPT-cases of Π-Induced Subgraph. Can we obtain a general lower bound for all FPT-cases of Π-Induced Subgraph? Interesting case: AT-free graphs.

Closed under embedding, so would fall under the first theorem. . . if only the Erd˝

s-Hajnal property was proven for them!
S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 11/12

SLIDE 74

Conclusions

We have shown kernelization hardness for most natural FPT-cases of Π-Induced Subgraph. Can we obtain a general lower bound for all FPT-cases of Π-Induced Subgraph? Interesting case: AT-free graphs.

Closed under embedding, so would fall under the first theorem. . . if only the Erd˝

s-Hajnal property was proven for them!

Big open problem: prove Erd˝

s-Hajnal conjecture.
S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 11/12

SLIDE 75

Thank you

Questions?

Tikz faces based on a code by Raoul Kessels, http://www.texample.net/tikz/examples/emoticons/, under Creative Commons Attribution 2.5 license (CC BY 2.5)

S. Kratsch, M. Pilipczuk, A. Rai, V. Raman

Kernel lower bounds using co-nondeterminism. . . 12/12