SDP Gaps for 2-to-1 and Other Label-Cover Variants

  • Venkatesan Guruswami
  • Subhash Khot
  • Ryan O’Donnell
  • Preyas Popat
  • Madhur Tulsiani
  • Yi Wu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)


In this paper we present semidefinite programming (SDP) gap instances for the following variants of the Label-Cover problem, closely related to the Unique Games Conjecture: (i) 2-to-1 Label-Cover; (ii) 2-to-2 Label-Cover; (iii) α-constraint Label-Cover. All of our gap instances have perfect SDP solutions. For alphabet size K, the integral optimal solutions have value: (i) \(O(1/\sqrt{\log K})\); (ii) O(1/logK); (iii) \(O(1/\sqrt{\log K})\).

Prior to this work, there were no known SDP gap instances for any of these problems with perfect SDP value and integral optimum tending to 0.


Alphabet Size Minimum Vertex Cover Inapproximability Result Label Cover Unique Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bulatov, A., Dalmau, V.: A simple algorithm for Mal’tsev constraints. SICOMP 36(1), 16–27 (2006)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Charikar, M., Makarychev, K., Makarychev, Y.: Near-optimal algorithms for unique games. In: Proc. 38th ACM STOC, pp. 205–214 (2006)Google Scholar
  3. 3.
    Dinur, I., Mossel, E., Regev, O.: Conditional hardness for approximate coloring. SICOMP 39(3), 843–873 (2009)MathSciNetGoogle Scholar
  4. 4.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162(1), 439–486 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Guruswami, V., Manokaran, R., Raghavendra, P.: Beating the random ordering is hard: Inapproximability of maximum acyclic subgraph. In: Proc. 49th IEEE FOCS, pp. 573–582 (2008)Google Scholar
  6. 6.
    Guruswami, V., Sinop, A.K.: Improved inapproximability results for maximum k-colorable subgraph. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009 and RANDOM 2009. LNCS, vol. 5687, pp. 163–176. Springer, Heidelberg (2009)Google Scholar
  7. 7.
    Khot, S.: Hardness results for coloring 3-colorable 3-uniform hypergraphs. In: Proc. 43rd IEEE FOCS, pp. 23–32 (2002)Google Scholar
  8. 8.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proc. 34th ACM STOC, pp. 767–775 (2002)Google Scholar
  9. 9.
    Khot, S., Saket, R.: SDP integrality gaps with local ℓ1-embeddability. In: Proc. 50th IEEE FOCS, pp. 565–574 (2009)Google Scholar
  10. 10.
    Khot, S., Vishnoi, N.: The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into ℓ1. In: Proc. 46th IEEE FOCS, pp. 53–62 (2005)Google Scholar
  11. 11.
    O’Donnell, R., Wu, Y.: Conditional hardness for satisfiable CSPs. In: Proc. 41st ACM STOC, pp. 493–502 (2009)Google Scholar
  12. 12.
    Raghavendra, P.: Optimal algorithms and inapproximability results for every CSP? In: Proc. 40th ACM STOC, pp. 245–254 (2008)Google Scholar
  13. 13.
    Raghavendra, P., Steurer, D.: Integrality gaps for strong SDP relaxations of unique games. In: Proc. 50th IEEE FOCS, pp. 575–585 (2009)Google Scholar
  14. 14.
    Schoenebeck, G.: Linear level Lasserre lower bounds for certain k-CSPs. In: Proc. 49th IEEE FOCS, pp. 593–602 (2008)Google Scholar
  15. 15.
    Talagrand, M.: On Russo’s approximate zero-one law. Ann. Prob. 22(3), 1576–1587 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Tang, L.: Conditional hardness of approximating satisfiable Max 3CSP-q. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 923–932. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Tulsiani, M.: CSP gaps and reductions in the Lasserre hierarchy. In: Proc. 41st ACM STOC, pp. 303–312 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  • Subhash Khot
    • 2
  • Ryan O’Donnell
    • 1
  • Preyas Popat
    • 2
  • Madhur Tulsiani
    • 3
  • Yi Wu
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon University 
  2. 2.Computer Science DepartmentNew York University 
  3. 3.School of MathematicsInstitute for Advanced Study 

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