, Volume 77, Issue 2, pp 555–594 | Cite as

Improved Approximation Algorithms for Projection Games

  • Pasin Manurangsi
  • Dana Moshkovitz


The projection games (aka Label Cover) problem is of great importance to the field of approximation algorithms, since most of the NP-hardness of approximation results we know today are reductions from Label Cover. In this paper we design several approximation algorithms for projection games: (1) A polynomial-time approximation algorithm that improves on the previous best approximation by Charikar et al. (Algorithmica 61(1):190–206, 2011). (2) A sub-exponential time algorithm with much tighter approximation for the case of smooth projection games. (3) A polynomial-time approximation scheme (PTAS) for projection games on planar graphs and a tight running time lower bound for such approximation schemes. The conference version of this paper had only the PTAS but not the running time lower bound.


Label Cover Projection games Approximation algorithms 


  1. 1.
    Arora, S., Barak, B., Steurer, D.: Subexponential algorithms for unique games and related problems. In: Proceedings of the 51st IEEE Symposium on Foundations of Computer Science, FOCS ’10, pp. 563–572. IEEE Computer Society, Washington, DC, USA (2010)Google Scholar
  2. 2.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM 45(3), 501–555 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. ACM 45(1), 70–122 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Babai, L., Fortnow, L., Levin, L.A., Szegedy, M.: Checking computations in polylogarithmic time. In: Proceedings of the 23rd ACM Symposium on Theory of Computing, STOC ’91, pp. 21–32. ACM, New York, NY, USA (1991)Google Scholar
  5. 5.
    Babai, L., Fortnow, L., Lund, C.: Non-deterministic exponential time has two-prover interactive protocols. Comput. Complex. 1(1), 3–40 (1991)CrossRefzbMATHGoogle Scholar
  6. 6.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bellare, M., Goldreich, O., Sudan, M.: Free bits, pcps, and nonapproximability-towards tight results. SIAM J. Comput. 27(3), 804–915 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Charikar, M., Hajiaghayi, M., Karloff, H.: Improved approximation algorithms for label cover problems. Algorithmica 61(1), 190–206 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dinur, I., Steurer, D.: Analytical approach to parallel repetition. In: Proceedings of the 46th ACM Symposium on Theory of Computing, STOC ’14, pp. 624–633. ACM, New York, NY, USA (2014)Google Scholar
  10. 10.
    Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27(3), 275–291 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Feige, U.: A threshold of \(\ln n\) for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Holmerin, J., Khot, S.: A new PCP outer verifier with applications to homogeneous linear equations and max-bisection. In: Proceedings of the 36th ACM Symposium on Theory of Computing, STOC ’04, pp. 11–20. ACM, New York, NY, USA (2004)Google Scholar
  15. 15.
    Khot, S.: Hardness results for coloring 3-colorable 3-uniform hypergraphs. In: Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, FOCS ’02, pp. 23–32. IEEE Computer Society, Washington, DC, USA (2002)Google Scholar
  16. 16.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the 34th ACM Symposium on Theory of Computing, STOC ’02, pp. 767–775. ACM, New York, NY, USA (2002)Google Scholar
  17. 17.
    Marx, D.: On the optimality of planar and geometric approximation schemes. In: Proceedings of the 48th IEEE Symposium on Foundations of Computer Science, FOCS ’07, pp. 338–348. IEEE Computer Society, Washington, DC, USA (2007)Google Scholar
  18. 18.
    Moshkovitz, D.: The projection games conjecture and the NP-hardness of \(\ln n\)-approximating set-cover. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques vol. 7408 of Lecture Notes in Computer Science, pp. 276–287. Springer, Berlin (2012)CrossRefGoogle Scholar
  19. 19.
    Moshkovitz, D., Raz, R.: Two-query PCP with subconstant error. J. ACM 57(5), 29:1–29:29 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Peleg, D.: Approximation algorithms for the label-cover max and red-blue set cover problems. J. Discret. Algorithms 5(1), 55–64 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Raz, R.: A parallel repetition theorem. SIAM J. Comput. 27(3), 763–803 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations