Abstract
This work considers the problem of approximating fixed predicate constraint satisfaction problems (MAX k-CSP(P)). We show that if the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on n variables cannot be approximated better than the trivial (random) approximation, even using Ω(n) levels of the Sherali-Adams LP hierarchy.
It was recently shown [3] that under the Unique Game Conjecture, CSPs with predicates with this condition cannot be approximated better than the trivial approximation. Our results can be viewed as an unconditional analogue of this result in the restricted computational model defined by the Sherali-Adams hierarchy. We also introduce a new generalization of techniques to define consistent “local distributions” over partial assignments to variables in the problem, which is often the crux of proving lower bounds for such hierarchies.
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References
Alekhnovich, M., Arora, S., Tourlakis, I.: Towards strong nonapproximability results in the Lovász-Schrijver hierarchy. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, May 22-24, 2005, ACM Press, New York (2005)
Arora, S., Bollobás, B., Lovász, L., Tourlakis, I.: Proving integrality gaps without knowing the linear program. Theory of Computing 2(2), 19–51 (2006)
Austrin, P., Mossel, E.: Approximation resistant predicates from pairwise independence. In: IEEE Conference on Computational Complexity, pp. 249–258. IEEE Computer Society Press, Los Alamitos (2008)
Bateni, M.H., Charikar, M., Guruswami, V.: MaxMin allocation via degree lower-bounded arborescences. In: STOC 2009. ACM Press, New York (2009)
Buresh-Oppenheim, J., Galesi, N., Hoory, S., Magen, A., Pitassi, T.: Rank bounds and integrality gaps for cutting planes procedures. Theory of Computing 2(4), 65–90 (2006)
Charikar, M., Makarychev, K., Makarychev, Y.: Integrality gaps for Sherali-Adams relaxations. In: STOC 2009. ACM Press, New York (2009)
Chlamtac, E.: Approximation algorithms using hierarchies of semidefinite programming relaxations. In: FOCS: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 691–701 (2007)
Chlamtac, E., Singh, G.: Improved approximation guarantees through higher levels of SDP hierarchies. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 49–62. Springer, Heidelberg (2008)
Engebretsen, L., Holmerin, J.: More efficient queries in pCPs for NP and improved approximation hardness of maximum CSP. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 194–205. Springer, Heidelberg (2005)
Feige, U., Krauthgamer, R.: The probable value of the Lovász-Schrijver relaxations for maximum independent set. SICOMP: SIAM Journal on Computing 32(2), 345–370 (2003)
de la Vega, W.F., Kenyon-Mathieu, C.: Linear programming relaxations of maxcut. In: SODA 2007: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, Philadelphia, PA, USA, pp. 53–61. Society for Industrial and Applied Mathematics (2007)
Georgiou, K., Magen, A., Pitassi, T., Tourlakis, I.: Integrality gaps of 2 − o(1) for Vertex Cover SDPs in the Lovász-Schrijver hierarchy. In: Proceedings of the 47th IEEE Symposium on Foundations of Computer Science, pp. 702–712 (2007)
Guruswami, V., Raghavendra, P.: Constraint satisfaction over a non-boolean domain: Approximation algorithms and unique-games hardness. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 77–90. Springer, Heidelberg (2008)
Lasserre, J.B.: An explicit exact SDP relaxation for nonlinear 0-1 programs. In: Aardal, K., Gerards, B. (eds.) IPCO 2001. LNCS, vol. 2081, pp. 293–303. Springer, Heidelberg (2001)
Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0-1 programming. Math. Oper. Res. 28(3), 470–496 (2003)
Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM Journal on Optimization 1(2), 166–190 (1991)
Magen, A., Moharrami, M.: Sherali-Adams based polynomial approximation schemes for NP-hard problems on planar and minor-free graphs (manuscript) (2008)
Mathieu, C., Sinclair, A.: Sherali-Adams relaxations of the matching polytope. In: STOC 2009. ACM Press, New York (2009)
Samorodnitsky, A., Trevisan, L.: A PCP characterization of NP with optimal amortized query complexity. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, STOC 2000, Portland, Oregon,, May 21-23, 2000, pp. 191–199. ACM Press, New York (2000)
Samorodnitsky, A., Trevisan, L.: Gowers uniformity, influence of variables, and PCPs. In: STOC 2006, pp. 11–20 (2006)
Schoenebeck, G.: Linear level lasserre lower bounds for certain k-CSPs. In: FOCS, pp. 593–602. IEEE Computer Society Press, Los Alamitos (2008)
Schoenebeck, G., Trevisan, L., Tulsiani, M.: A linear round lower bound for Lovász-Schrijver SDP relaxations of vertex cover. In: IEEE Conference on Computational Complexity, pp. 205–216. IEEE Computer Society Press, Los Alamitos (2007)
Schoenebeck, G., Trevisan, L., Tulsiani, M.: Tight integrality gaps for Lovász-Schrijver LP relaxations of vertex cover and max cut. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007. ACM Press, New York (2007)
Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3(3), 411–430 (1990)
Tourlakis, I.: Towards optimal integrality gaps for hypergraph vertex cover in the Lovász-Schrijver hierarchy. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 233–244. Springer, Heidelberg (2005)
Tourlakis, I.: New lower bounds for vertex cover in the Lovász-Schrijver hierarchy. In: Proceedings of the 21st IEEE Conference on Computational Complexity, pp. 170–182. IEEE Computer Society Press, Los Alamitos (2006)
Tulsiani, M.: CSP gaps and reductions in the Lasserre hierarchy. In: STOC 2009. ACM Press, New York (2009)
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Georgiou, K., Magen, A., Tulsiani, M. (2009). Optimal Sherali-Adams Gaps from Pairwise Independence. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_10
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DOI: https://doi.org/10.1007/978-3-642-03685-9_10
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