Article

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

Authors:

Stefan Kratsch,

Marcin Pilipczuk,

Venkatesh RamanAuthors Info & Claims

SWAT'12: Proceedings of the 13th Scandinavian conference on Algorithm Theory

Pages 364 - 375

https://doi.org/10.1007/978-3-642-31155-0_32

Published: 04 July 2012 Publication History

Abstract

This work further explores the applications of co-nondeterminism for showing kernelization lower bounds. The only known example excludes polynomial kernelizations for the Ramsey problem of finding an independent set or a clique of at least k vertices in a given graph (Kratsch 2012, SODA). We study the more general problem of finding induced subgraphs on k vertices fulfilling some hereditary property Π, called Π-Induced Subgraph. The problem is NP-hard for all non-trivial choices of Π by a classic result of Lewis and Yannakakis (JCSS 1980). The parameterized complexity of this problem was classified by Khot and Raman (TCS 2002) depending on the choice of Π. The interesting cases for kernelization are for Π containing all independent sets and all cliques, since the problem is trivial or W[1]-hard otherwise.

Our results are twofold. Regarding Π-Induced Subgraph, we show that for a large choice of natural graph properties Π, including chordal, perfect, cluster, and cograph, there is no polynomial kernel with respect to k. This is established by two theorems: one using a co-nondeterministic variant of cross-composition and one by a polynomial parameter transformation from Ramsey.

Additionally, we show how to use improvement versions of NP-hard problems as source problems for lower bounds, without requiring their NP-hardness. E.g., for Π-Induced Subgraph our compositions may assume existing solutions of size k−1. We believe this to be useful for further lower bound proofs, since improvement versions simplify the construction of a disjunction (OR) of instances required in compositions. This adds a second way of using co-nondeterminism for lower bounds.

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Cited By

Jansen BMarx DIndyk P(2015)Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernelsProceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms10.5555/2722129.2722171(616-629)Online publication date: 4-Jan-2015
https://dl.acm.org/doi/10.5555/2722129.2722171
Kratsch SPilipczuk MRai ARaman V(2015)Kernel Lower Bounds using Co-Nondeterminism: Finding Induced Hereditary SubgraphsACM Transactions on Computation Theory10.1145/26913217:1(1-18)Online publication date: 13-Jan-2015
https://dl.acm.org/doi/10.1145/2691321
Fellows MJansen B(2014)FPT is characterized by useful obstruction setsACM Transactions on Computation Theory10.1145/26358206:4(1-26)Online publication date: 1-Aug-2014
https://dl.acm.org/doi/10.1145/2635820
Show More Cited By

Kernel lower bounds using co-nondeterminism: finding induced hereditary subgraphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
2. Theory of computation

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cover image Guide Proceedings

SWAT'12: Proceedings of the 13th Scandinavian conference on Algorithm Theory

July 2012

397 pages

ISBN:9783642311543

Editors:
Fedor V. Fomin
Institute of Informatics, University of Bergen, Bergen, Norway
,
Petteri Kaski
Department of Information and Computer Science, Aalto University, Aalto, Finland

Sponsors

University of Helsinki
Federation of Finnish Learned Societies
Aalto University

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 04 July 2012

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Cited By

Jansen BMarx DIndyk P(2015)Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernelsProceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms10.5555/2722129.2722171(616-629)Online publication date: 4-Jan-2015
https://dl.acm.org/doi/10.5555/2722129.2722171
Kratsch SPilipczuk MRai ARaman V(2015)Kernel Lower Bounds using Co-Nondeterminism: Finding Induced Hereditary SubgraphsACM Transactions on Computation Theory10.1145/26913217:1(1-18)Online publication date: 13-Jan-2015
https://dl.acm.org/doi/10.1145/2691321
Fellows MJansen B(2014)FPT is characterized by useful obstruction setsACM Transactions on Computation Theory10.1145/26358206:4(1-26)Online publication date: 1-Aug-2014
https://dl.acm.org/doi/10.1145/2635820
Kratsch S(2014)Co-Nondeterminism in CompositionsACM Transactions on Algorithms10.1145/263580810:4(1-16)Online publication date: 11-Aug-2014
https://dl.acm.org/doi/10.1145/2635808

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