skip to main content
research-article

Kernel Lower Bounds using Co-Nondeterminism: Finding Induced Hereditary Subgraphs

Published:13 January 2015Publication History
Skip Abstract Section

Abstract

This work further explores the applications of co-nondeterminism for showing kernelization lower bounds. The only known example prior to this work excludes polynomial kernelizations for the so-called Ramsey problem of finding an independent set or a clique of at least k vertices in a given graph [Kratsch 2012]. 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 nontrivial choices of Π by a classic result of Lewis and Yannakakis [1980]. The parameterized complexity of this problem was classified by Khot and Raman [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 trivially polynomial time solvable 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, each one capturing different (but not necessarily exclusive) sets of properties: one using a co-nondeterministic variant of OR-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. For example, for Π-Induced Subgraph our compositions may assume existing solutions of size k--1. This follows from the more general fact that source problems for OR-(cross-)compositions need only be NP-hard under co-nondeterministic reductions. We believe this to be useful for further lower-bound proofs, for example, 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.

References

  1. Noga Alon, János Pach, and József Solymosi. 2001. Ramsey-type theorems with forbidden subgraphs. Combinatorica 21, 2, 155--170.Google ScholarGoogle Scholar
  2. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. 2009. On problems without polynomial kernels. J. Comput. System Sci. 75, 8, 423--434. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. 2014. Kernelization lower bounds by cross-composition. SIAM J. Disc. Math. 28, 1, 277--305.Google ScholarGoogle ScholarCross RefCross Ref
  4. Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. 2011. Kernel bounds for disjoint cycles and disjoint paths. Theoret. Comput. Sci. 412, 35, 4570--4578. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Leizhen Cai. 1996. Fixed-parameter tractability of graph modification problems for hereditary properties. Inform. Process. Lett. 58, 171--176. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Holger Dell and Dániel Marx. 2012. Kernelization of packing problems. In Proceedings of SODA. 68--81. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Holger Dell and Dieter van Melkebeek. 2010. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In Proceedings of STOC. 251--260. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Reinhard Diestel. 2005. Graph Theory. Springer.Google ScholarGoogle Scholar
  9. Michael Dom, Daniel Lokshtanov, and Saket Saurabh. 2009. Incompressibility through colors and IDs. In Proceedings of ICALP (1). 378--389. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Paul Erdős. 1947. Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53, 292--294.Google ScholarGoogle ScholarCross RefCross Ref
  11. Paul Erdős and András Hajnal. 1989. Ramsey-type theorems. Disc. Appl. Math. 25, 1--2, 37--52. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Paul Erdős and George Szekeres. 1935. A combinatorial problem in geometry. Compositio Mathematica 2, 463--470.Google ScholarGoogle Scholar
  13. Fedor V. Fomin, Saket Saurabh, and Yngve Villanger. 2013. A polynomial kernel for proper interval vertex deletion. SIAM J. Disc. Math. 27, 4, 1964--1976.Google ScholarGoogle ScholarCross RefCross Ref
  14. Lance Fortnow and Rahul Santhanam. 2011. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. System Sci. 77, 1, 91--106. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Martin C. Golumbic. 2004. Algorithmic Graph Theory and Perfect Graphs, 2nd Ed. Elsevier Science. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Danny Harnik and Moni Naor. 2010. On the compressibility of NP instances and cryptographic applications. SIAM J. Comput. 39, 5, 1667--1713. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Danny Hermelin and Xi Wu. 2012. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proceedings of SODA. 104--113. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Subhash Khot and Venkatesh Raman. 2002. Parameterized complexity of finding subgraphs with hereditary properties. Theor. Computer Science 289, 2, 997--1008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Stefan Kratsch. 2012. Co-nondeterminism in compositions: A kernelization lower bound for a Ramsey-type problem. In Proceedings of SODA. 114--122. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Stefan Kratsch, Marcin Pilipczuk, Ashutosh Rai, and Venkatesh Raman. 2012. Kernel lower bounds using co-nondeterminism: Finding induced hereditary subgraphs. In Proceedings of SWAT. 364--375. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. John M. Lewis and Mihalis Yannakakis. 1980. The node-deletion problem for hereditary properties is NP-complete. J. Comput. System Sci. 20, 2, 219--230.Google ScholarGoogle ScholarCross RefCross Ref
  22. László Lovasz. 1983. Perfect graphs. In Selected Topics in Graph Theory, Volume 2, L. W. Beineke and R. J. Wilson (Eds.), Academic Press, London-New York, 55--67.Google ScholarGoogle Scholar
  23. Dániel Marx. 2010. Chordal deletion is fixed-parameter tractable. Algorithmica 57, 4, 747--768. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Dániel Marx and Ildikó Schlotter. 2012. Obtaining a planar graph by vertex deletion. Algorithmica 62, 3--4, 807--822. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. René van Bevern, Christian Komusiewicz, Hannes Moser, and Rolf Niedermeier. 2010. Measuring indifference: Unit interval vertex deletion. In Proceedings of WG. 232--243. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Pim van ’t Hof and Yngve Villanger. 2013. Proper interval vertex deletion. Algorithmica 65, 4, 845--867.Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Kernel Lower Bounds using Co-Nondeterminism: Finding Induced Hereditary Subgraphs

          Recommendations

          Comments

          Login options

          Check if you have access through your login credentials or your institution to get full access on this article.

          Sign in

          Full Access

          • Published in

            cover image ACM Transactions on Computation Theory
            ACM Transactions on Computation Theory  Volume 7, Issue 1
            December 2014
            104 pages
            ISSN:1942-3454
            EISSN:1942-3462
            DOI:10.1145/2692372
            Issue’s Table of Contents

            Copyright © 2015 ACM

            Publisher

            Association for Computing Machinery

            New York, NY, United States

            Publication History

            • Published: 13 January 2015
            • Accepted: 1 June 2014
            • Revised: 1 May 2014
            • Received: 1 October 2013
            Published in toct Volume 7, Issue 1

            Permissions

            Request permissions about this article.

            Request Permissions

            Check for updates

            Qualifiers

            • research-article
            • Research
            • Refereed

          PDF Format

          View or Download as a PDF file.

          PDF

          eReader

          View online with eReader.

          eReader
          About Cookies On This Site

          We use cookies to ensure that we give you the best experience on our website.

          Learn more

          Got it!