Skip to main content
SpringerLink
Log in

On Polynomial Kernels for Integer Linear Programs: Covering, Packing and Feasibility

  • Conference paper
Algorithms – ESA 2013 (ESA 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8125))

Included in the following conference series:

Abstract

We study the existence of polynomial kernels for the problem of deciding feasibility of integer linear programs (ILPs), and for finding good solutions for covering and packing ILPs. Our main results are as follows: First, we show that the ILP Feasibility problem admits no polynomial kernelization when parameterized by both the number of variables and the number of constraints, unless NP ⊆ coNP/poly. This extends to the restricted cases of bounded variable degree and bounded number of variables per constraint, and to covering and packing ILPs. Second, we give a polynomial kernelization for the Cover ILP problem, asking for a solution to Ax ≥ b with c T x ≤ k, parameterized by k, when A is row-sparse; this generalizes a known polynomial kernelization for the special case with 0/1-variables and coefficients (d-Hitting Set).

Some proofs are omitted from this extended abstract and can be found in [1].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Kratsch, S.: On polynomial kernels for integer linear programs: Covering, packing and feasibility. CoRR abs/1302.3496 (2013)

    Google Scholar 

  2. Atamtürk, A., Savelsbergh, M.W.P.: Integer-programming software systems. Annals OR 140(1), 67–124 (2005)

    Article  MATH  Google Scholar 

  3. Harnik, D., Naor, M.: On the compressibility of \(\mathcal{NP}\) instances and cryptographic applications. SIAM J. Comput. 39(5), 1667–1713 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer (1998)

    Google Scholar 

  5. Lokshtanov, D., Misra, N., Saurabh, S.: Kernelization – preprocessing with a guarantee. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) Fellows Festschrift. LNCS, vol. 7370, pp. 129–161. Springer, Heidelberg (2012)

    Google Scholar 

  6. Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75, 423–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lenstra, H.W.: Integer programming with a fixed number of variables. Mathematics of Operations Research 8, 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kannan, R.: Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research 12(3), 415–440 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and ids. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  11. Hermelin, D., Kratsch, S., Soltys, K., Wahlström, M., Wu, X.: Hierarchies of inefficient kernelizability. CoRR abs/1110.0976 (2011)

    Google Scholar 

  12. Kratsch, S.: On polynomial kernels for sparse integer linear programs. In: Portier, N., Wilke, T. (eds.) STACS. LIPIcs, vol. 20, pp. 80–91. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)

    Google Scholar 

  13. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: On completeness for W[1]. Theor. Comput. Sci. 141(1&2), 109–131 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jansen, B.M.P., Bodlaender, H.L.: Vertex cover kernelization revisited - upper and lower bounds for a refined parameter. Theory Comput. Syst. 53(2), 263–299 (2013)

    Article  MathSciNet  Google Scholar 

  15. Nederlof, J., van Leeuwen, E.J., van der Zwaan, R.: Reducing a target interval to a few exact queries. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 718–727. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  16. Plotkin, S.A., Shmoys, D.B., Tardos, É.: Fast approximation algorithms for fractional packing and covering problems. In: FOCS, pp. 495–504 (1991)

    Google Scholar 

  17. Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer (2006)

    Google Scholar 

  18. Abu-Khzam, F.N.: A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Schwentick, T., Dürr, C. (eds.) STACS. LIPIcs, vol. 9, pp. 165–176 (2011)

    Google Scholar 

  20. Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. London Math. Soc. 35, 85–90 (1960)

    Article  MathSciNet  Google Scholar 

  21. Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technical University Berlin, Germany

    Stefan Kratsch

Authors
  1. Stefan Kratsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Utrecht University, Princetonplein 5, 3584 CC, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma “Tor Vergata”, via del Politecnico 1, 00133, Rome, Italy

    Giuseppe F. Italiano

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kratsch, S. (2013). On Polynomial Kernels for Integer Linear Programs: Covering, Packing and Feasibility. In: Bodlaender, H.L., Italiano, G.F. (eds) Algorithms – ESA 2013. ESA 2013. Lecture Notes in Computer Science, vol 8125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40450-4_55

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40450-4_55

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40449-8

  • Online ISBN: 978-3-642-40450-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation