# Improved Approximation Algorithms for Projection Games

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## Abstract

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.

## Keywords

Label Cover Projection games Approximation algorithms## References

- 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.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)MathSciNetCrossRefMATHGoogle Scholar - 3.Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. ACM
**45**(1), 70–122 (1998)MathSciNetCrossRefMATHGoogle Scholar - 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.Babai, L., Fortnow, L., Lund, C.: Non-deterministic exponential time has two-prover interactive protocols. Comput. Complex.
**1**(1), 3–40 (1991)CrossRefMATHGoogle Scholar - 6.Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM
**41**(1), 153–180 (1994)MathSciNetCrossRefMATHGoogle Scholar - 7.Bellare, M., Goldreich, O., Sudan, M.: Free bits, pcps, and nonapproximability-towards tight results. SIAM J. Comput.
**27**(3), 804–915 (1998)MathSciNetCrossRefMATHGoogle Scholar - 8.Charikar, M., Hajiaghayi, M., Karloff, H.: Improved approximation algorithms for label cover problems. Algorithmica
**61**(1), 190–206 (2011)MathSciNetCrossRefMATHGoogle Scholar - 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.Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica
**27**(3), 275–291 (2000)MathSciNetCrossRefMATHGoogle Scholar - 11.Feige, U.: A threshold of \(\ln n\) for approximating set cover. J. ACM
**45**(4), 634–652 (1998)MathSciNetCrossRefMATHGoogle Scholar - 12.Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci.
**1**(3), 237–267 (1976)MathSciNetCrossRefMATHGoogle Scholar - 13.Håstad, J.: Some optimal inapproximability results. J. ACM
**48**(4), 798–859 (2001)MathSciNetCrossRefMATHGoogle Scholar - 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.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.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.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.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.Moshkovitz, D., Raz, R.: Two-query PCP with subconstant error. J. ACM
**57**(5), 29:1–29:29 (2008)MathSciNetMATHGoogle Scholar - 20.Peleg, D.: Approximation algorithms for the label-cover max and red-blue set cover problems. J. Discret. Algorithms
**5**(1), 55–64 (2007)MathSciNetCrossRefMATHGoogle Scholar - 21.Raz, R.: A parallel repetition theorem. SIAM J. Comput.
**27**(3), 763–803 (1998)MathSciNetCrossRefMATHGoogle Scholar