Abstract
Lovász and Schrijver described a generic method of tightening the LP and SDP relaxation for any 0–1 optimization problem. These tightened relaxations were the basis of several celebrated approximation algorithms (such as for max-cut, max-3sat, and sparsest cut).
We prove strong inapproximability results in this model for well-known problems such as max-3sat, hypergraph vertex cover and set cover. We show that the relaxations produced by as many as Ω(n) rounds of the LS + procedure do not allow nontrivial approximation, thus ruling out the possibility that the LS + approach gives even slightly subexponential approximation algorithms for these problems.
We also point out why our results are somewhat incomparable to known inapproximability results proved using PCPs, and formalize several interesting open questions. We survey results that built upon this work.
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References
Sanjeev Arora, Béla Bollobás & László Lovász (2002). Proving Integrality Gaps without Knowing the Linear Program. In Proceedings of the 43rd Symposium on Foundations of Computer Science (FOCS-02), 313–322. Los Alamitos.
Sanjeev Arora, Béla Bollobás, László Lovász & Iannis Tourlakis (2006). Proving Integrality Gaps without Knowing the Linear Program. Theory of Computing 2(2), 19–51. URL http://www.theoryofcomputing.org/articles/main/v002/a002.
Sanjeev Arora, Satish Rao & Umesh V. Vazirani (2009). Expander flows, geometric embeddings and graph partitioning. J. ACM 56(2). Preliminary version in ACM STOC 2004.
Joshua Buresh-Oppenheim, Nicola Galesi, Shlomo Hoory, Avner Magen & Toniann Pitassi (2006). Rank Bounds and Integrality Gaps for Cutting Planes Procedures. Theory of Computing 2(4), 65–90. URL http://www.theoryofcomputing.org/articles/main/v002/a004.
Moses Charikar, Konstantin Makarychev & Yury Makarychev (2009). Integrality gaps for Sherali-Adams relaxations. In Proc. ACM STOC, 283–292.
Irit Dinur, Venkatesan Guruswami, Subhash Khot & Oded Regev (2005). A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover. SIAM J. Comput. 34(5), 1129–1146. Preliminary Version in ACM STOC 2003.
Dinur Irit, Safra Shmuel (2005) On the hardness of approximating minimum vertex-cover. Annals of Mathematics 162(1): 439–486
Feige Uriel (1998) A Threshold of ln for Approximating Set Cover. J. ACM 45(4): 634–652
Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra & Mario Szegedy (1996). Interactive Proofs and the Hardness of Approximating Cliques. J. ACM 43(2), 268–292. URL http://doi.acm.org/10.1145/226643.226652.
Uriel Feige, Jeong Han Kim & Eran Ofek (2006). Witnesses for non-satisfiability of dense random 3CNF formulas. In FOCS, 497–508. IEEE Computer Society. ISBN 0-7695-2720-5. URL http://doi.ieeecomputersociety.org/10.1109/FOCS.2006.78.
Uriel Feige & Robert Krauthgamer (2003). The Probable Value of the Lovász–Schrijver Relaxations for Maximum Independent Set. SIAM Journal on Computing 32(2), 345–370. ISSN 0097-5397 (print), 1095-7111 (electronic). URL http://epubs.siam.org/sam-bin/dbq/article/40118.
Uriel Feige & Eran Ofek (2007). Easily refutable subformulas of large random 3CNF formulas. Theory of Computing 3(2), 25–43. URL http://www.theoryofcomputing.org/articles/main/v003/a002.
Konstantinos Georgiou, Avner Magen, Toniann Pitassi & Iannis Tourlakis (2007). Integrality gaps of 2−o(1) for vertex cover SDPs in the Lovász-Schrijver hierarchy. In Manuscript.
Goemans Michel X., Tunçel Levent (2001) When Does the Positive Semidefiniteness Constraint Help In Lifting Procedures. Mathematics of Operations Research 26: 796–815
Goemans Michel X., Williamson David P. (1995) Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42(6): 1115–1145 ISSN 0004-5411
Håstad Johan (2001) Some optimal inapproximability results. J. ACM 48(4): 798–859
Subash Khot (2002). On the power of unique 2-prover 1-round games. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, 767–775 (electronic). ACM, New York.
László Lovász & Alexander Schrijver (1991). Cones of matrices and set-functions and 0–1 optimization. SIAM Journal on Optimization 1(2), 166–190. ISSN 1052-6234 (print), 1095-7189 (electronic).
Prasad Raghavendra & David Steurer (2009). Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES. In Proc. IEEE FOCS, 575–585.
Ran Raz & Shmuel Safra (1997). A Sub-Constant Error-Probability Low-Degree Test, and a Sub-Constant Error-Probability PCP Characterization of NP. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, 475–484. El Paso, Texas.
Grant Schoenebeck (2008). Linear Level Lasserre Lower Bounds for Certain k-CSPs. In FOCS ’08: Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, 593–602. IEEE Computer Society, Washington, DC, USA. ISBN 978-0-7695-3436-7.
Grant Schoenebeck, Luca Trevisan & Madhur Tulsiani (2006). A linear round lower bound for lovasz-schrijver sdp relaxations of vertex cover. In In IEEE Conference on Computational Complexity. IEEE Computer Society, 06–098.
Grant Schoenebeck, Luca Trevisan & Madhur Tulsiani (2007). Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut. In Proc. ACM STOC, 302–310.
Sherali Hanif D., Adams Warren P. (1990) A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems. SIAM Journal on Discrete Mathematics 3(3): 411–430 ISSN 0895-4801 (print), 1095-7146 (electronic)
Iannis Tourlakis (2005). Towards Optimal Integrality Gaps for Hypergraph Vertex Cover in the Lovász-Schrijver Hierarchy. In 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX, 233–244.
Wenceslas Fernandez de la Vega & Claire Kenyon-Mathieu (2007). Linear Programming Relaxations of Maxcut. In Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms.
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Alekhnovich, M., Arora, S. & Tourlakis, I. Towards Strong Nonapproximability Results in the Lovász-Schrijver Hierarchy. comput. complex. 20, 615–648 (2011). https://doi.org/10.1007/s00037-011-0027-z
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DOI: https://doi.org/10.1007/s00037-011-0027-z
Keywords
- Algorithms
- NP-complete problems
- approximation
- proof complexity
- lift and project methods
- matrix cut operators
- lower bounds
- random instances of 3SAT