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

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

Kernelization instance of NP-hard problem S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 2/12

Kernelization k instance of NP-hard problem S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 2/12

Kernelization k instance of NP-hard problem S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 2/12

Kernelization k poly time instance of NP-hard problem S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 2/12

Kernelization k poly time size ≤ g ( k ) instance of NP-hard problem S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 2/12

Kernelization k poly time size ≤ g ( k ) instance of NP-hard problem small: g ( k ) polynomial S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 2/12

Kernelization k poly time size ≤ g ( k ) instance of NP-hard problem small: g ( k ) polynomial sometimes impossible S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 2/12

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

Kernelization lower bounds [Bodlaender, Downey, Fellows, Hermelin, ICALP’08] t instances of hard language L S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 3/12

Kernelization lower bounds [Bodlaender, Downey, Fellows, Hermelin, ICALP’08] poly time t instances of hard language L S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 3/12

Kernelization lower bounds [Bodlaender, Downey, Fellows, Hermelin, ICALP’08] k poly time t instances OR instance of hard language L of param. lang. Q S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 3/12

Kernelization lower bounds [Bodlaender, Downey, Fellows, Hermelin, ICALP’08] k poly time poly time t instances OR instance of hard language L of param. lang. Q S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 3/12

Kernelization lower bounds [Bodlaender, Downey, Fellows, Hermelin, ICALP’08] k poly time poly time size ≤ poly ( k ) t instances OR instance of hard language L of param. lang. Q S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 3/12

Kernelization lower bounds [Bodlaender, Downey, Fellows, Hermelin, ICALP’08] k poly time poly time size ≤ poly ( k ) t instances OR instance of hard language L of param. lang. Q [Fortnow, Santhanam, STOC’08] ⇒ L ∈ coNP / poly S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 3/12

Kernelization lower bounds [Bodlaender, Downey, Fellows, Hermelin, ICALP’08] k poly time poly time size ≤ poly ( k ) t instances OR instance of hard language L of param. lang. Q [Fortnow, Santhanam, STOC’08] ⇒ L ∈ coNP / poly L is NP-hard ⇒ NP ⊆ coNP / poly ⇒ PH = Σ P 3 S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 3/12

Kernelization lower bounds [Bodlaender, Downey, Fellows, Hermelin, ICALP’08] k coNP time coNP time size ≤ poly ( k ) t instances OR instance of hard language L of param. lang. Q [Fortnow, Santhanam, STOC’08] ⇒ L ∈ coNP / poly L is NP-hard ⇒ NP ⊆ coNP / poly ⇒ PH = Σ P 3 S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 3/12

Kernelization lower bounds [Bodlaender, Downey, Fellows, Hermelin, ICALP’08] k coNP time coNP time size ≤ poly ( k ) t instances OR instance of hard language L of param. lang. Q [Fortnow, Santhanam, STOC’08] ⇒ L ∈ coNP / poly L is NP-hard ⇒ NP ⊆ coNP / poly ⇒ PH = Σ P 3 under “coNP reductions” S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 3/12

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

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

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

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

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 x i in co-nondeterministic poly-time into one OR-instance of Q with parameter ≤ poly (max i | x i | ) t o (1) . S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 4/12

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 x i in co-nondeterministic poly-time into one OR-instance of Q with parameter ≤ poly (max i | x i | ) t o (1) . That is, if at least one x i ∈ L , then all outputs ∈ Q . S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using co-nondeterminism. . . 4/12

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 x i in co-nondeterministic poly-time into one OR-instance of Q with parameter ≤ poly (max i | x i | ) t o (1) . That is, if at least one x i ∈ L , then all outputs ∈ Q . If all x i / ∈ 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

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 x i in co-nondeterministic poly-time into one OR-instance of Q with parameter ≤ poly (max i | x i | ) t o (1) . That is, if at least one x i ∈ L , then all outputs ∈ Q . If all x i / ∈ 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

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

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

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