Skip to main content
SpringerLink
Log in

An Improved Integrality Gap for the Călinescu-Karloff-Rabani Relaxation for Multiway Cut

  • Conference paper
  • First Online:
Integer Programming and Combinatorial Optimization (IPCO 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10328))

Abstract

We construct an improved integrality gap instance for the Călinescu-Karloff-Rabani LP relaxation of the Multiway Cut problem. For \(k \geqslant 3\) terminals, our instance has an integrality ratio of \(6 / (5 + \frac{1}{k - 1}) - \varepsilon \), for every constant \(\varepsilon > 0\). For every \(k \geqslant 4\), this improves upon a long-standing lower bound of \(8 / (7 + \frac{1}{k - 1})\) by Freund and Karloff [7]. Due to the result by Manokaran et al. [9], our integrality gap also implies Unique Games hardness of approximating Multiway Cut of the same ratio.

Y. Makarychev—Supported by NSF awards CAREER CCF-1150062 and IIS-1302662.

P. Manurangsi—Supported by NSF Grants No. CCF-1540685 and CCF-1655215. Part of this work was done while the author was visiting TTIC.

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 EPUB and 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

Notes

  1. 1.

    In [8], \(\tau _k^*\) is defined as the infimum of \(\tau _k(\mathcal {P})\) among all \(\mathcal {P}\)’s but it was proved in the same work that there exists \(\mathcal {P}\) that achieves the infimum.

  2. 2.

    The minimum exists since there is only a finite number of k-way cuts of the discretized simplex \(\varDelta _{k, n}\).

  3. 3.

    This terminology is from [5] but our instance differs significantly from theirs.

References

  1. Angelidakis, H., Makarychev, Y., Manurangsi, P.: An improved integrality gap for the Calinescu-Karloff-Rabani relaxation for multiway cut. CoRR abs/1611.05530 (2016). http://arxiv.org/abs/1611.05530

  2. Buchbinder, N., Naor, J., Schwartz, R.: Simplex partitioning via exponential clocks and the multiway cut problem. In: Proceedings of the 45th ACM Symposium on Theory of Computing, STOC, pp. 535–544 (2013)

    Google Scholar 

  3. Buchbinder, N., Schwartz, R., Weizman, B.: Simplex transformations and the multiway cut problem. In: Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 2400–2410 (2017)

    Google Scholar 

  4. Călinescu, G., Karloff, H.J., Rabani, Y.: An improved approximation algorithm for multiway cut. J. Comput. Syst. Sci. 60(3), 564–574 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cunningham, W.H., Tang, L.: Optimal 3-terminal cuts and linear programming. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, pp. 114–125. Springer, Heidelberg (1999). doi:10.1007/3-540-48777-8_9

    Chapter  Google Scholar 

  6. Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23(4), 864–894 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Freund, A., Karloff, H.: A lower bound of \(8/(7 + \frac{1}{k - 1})\) on the integrality ratio of the Călinescu-Karloff-Rabani relaxation for multiway cut. Inf. Process. Lett. 75(1–2), 43–50 (2000)

    Article  MATH  Google Scholar 

  8. Karger, D.R., Klein, P.N., Stein, C., Thorup, M., Young, N.E.: Rounding algorithms for a geometric embedding of minimum multiway cut. Math. Oper. Res. 29(3), 436–461 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Manokaran, R., Naor, J., Raghavendra, P., Schwartz, R.: SDP gaps and UGC hardness for multiway cut, 0-extension, and metric labeling. In: Proceedings of the 40th ACM Symposium on Theory of Computing, STOC, pp. 11–20 (2008)

    Google Scholar 

  10. Sharma, A., Vondrák, J.: Multiway cut, pairwise realizable distributions, and descending thresholds. In: Proceedings of the 46th ACM Symposium on Theory of Computing, STOC, pp. 724–733 (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Toyota Technological Institute at Chicago, Chicago, IL, USA

    Haris Angelidakis & Yury Makarychev

  2. University of California, Berkeley, Berkeley, CA, USA

    Pasin Manurangsi

Authors
  1. Haris Angelidakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yury Makarychev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pasin Manurangsi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pasin Manurangsi .

Editor information

Editors and Affiliations

  1. Ecole Polytechnique Federale de Lausanne - EPFL, Lausanne, Switzerland

    Friedrich Eisenbrand

  2. University of Waterloo, Waterloo, Ontario, Canada

    Jochen Koenemann

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Angelidakis, H., Makarychev, Y., Manurangsi, P. (2017). An Improved Integrality Gap for the Călinescu-Karloff-Rabani Relaxation for Multiway Cut. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-59250-3_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59249-7

  • Online ISBN: 978-3-319-59250-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation