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

  • Haris Angelidakis
  • Yury Makarychev
  • Pasin ManurangsiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)


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.


  1. 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).
  2. 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. 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. 4.
    Călinescu, G., Karloff, H.J., Rabani, Y.: An improved approximation algorithm for multiway cut. J. Comput. Syst. Sci. 60(3), 564–574 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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 CrossRefGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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)CrossRefzbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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. 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

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Haris Angelidakis
    • 1
  • Yury Makarychev
    • 1
  • Pasin Manurangsi
    • 2
    Email author
  1. 1.Toyota Technological Institute at ChicagoChicagoUSA
  2. 2.University of California, BerkeleyBerkeleyUSA

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