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Strong reductions for extended formulations

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Abstract

We generalize the reduction mechanism for linear programming problems and semidefinite programming problems from Braun et al. (Inapproximability of combinatorial problems via small LPs and SDPs, 2015) in two ways (1) relaxing the requirement of affineness, and (2) extending to fractional optimization problems. As applications we provide several new LP-hardness and SDP-hardness results, e.g., for the problem, the problem, and the problem and show how to extend ad-hoc reductions between Sherali–Adams relaxations to reductions between LPs.

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References

  1. Arora, S., Lee, J., Naor, A.: Euclidean distortion and the sparsest cut. J. Am. Math. Soc. 21(1), 1–21 (2008)

    Article  MathSciNet  Google Scholar 

  2. Bansal, N., Khot, S.: Optimal long code test with one free bit. In: Proceedings of FOCS. IEEE (2009). https://doi.org/10.1109/FOCS.2009.23

  3. Bazzi, A., Fiorini, S., Pokutta, S., Svensson, O.: No small linear program approximates Vertex Cover within a factor \(2-\varepsilon \). arXiv preprint arXiv:1503.00753 (2015)

  4. Braun, G., Fiorini, S., Pokutta, S., Steurer, D.: Approximation limits of linear programs (beyond hierarchies). In: 53rd IEEE Symposium on Foundations of Computer Science (FOCS 2012), pp. 480–489 (2012). https://doi.org/10.1109/FOCS.2012.10

  5. Braun, G., Pokutta, S., Zink, D.: Inapproximability of combinatorial problems via small LPs and SDPs. In: Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing (STOC), pp. 107–116. ACM (2015)

  6. Chan, S.O., Lee, J.R., Raghavendra, P., Steurer, D.: Approximate constraint satisfaction requires large LP relaxations. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pp. 350–359. IEEE (2013)

  7. Charikar, M., Makarychev, K., Makarychev, Y.: Integrality gaps for Sherali–Adams relaxations. In: Proceedings of the Forty-First Annual ACM Symposium on Theory of computing, pp. 283–292. ACM (2009)

  8. Chawla, S., Krauthgamer, R., Kumar, R., Rabani, Y., Sivakumar, D.: On the hardness of approximating multicut and sparsest-cut. Comput. Complex. 15(2), 94–114 (2006)

    Article  MathSciNet  Google Scholar 

  9. Cheeger, J., Kleiner, B., Naor, A.: A \((\log n)^{\Omega (1)}\) integrality gap for the sparsest cut SDP. In: 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS’09), pp. 555–564. IEEE (2009)

  10. Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: Exponential separation and strong lower bounds. In: Proceedings of STOC, pp. 95–106 (2012)

  11. Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing (STOC), pp. 95–106. ACM (2012)

  12. Gupta, A., Talwar, K., Witmer, D.: Sparsest cut on bounded treewidth graphs: algorithms and hardness results. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of computing, pp. 281–290. ACM (2013)

  13. Khot, S.A., Vishnoi, N.K.: The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative-type metrics into \(\ell _{1}\). J. ACM (JACM) 62(1), 8 (2015)

    Article  Google Scholar 

  14. Kothari, P.K., Meka, R., Raghavendra, P.: Approximating rectangles by juntas and weakly-exponential lower bounds for LP relaxations of CSPs. In: Proceedings of STOC, pp. 590–603. ACM, New York (2017). https://doi.org/10.1145/3055399.3055438

  15. Lee, J.R., Raghavendra, P., Steurer, D.: Lower bounds on the size of semidefinite programming relaxations. arXiv preprint arXiv:1411.6317 (2014)

  16. Pashkovich, K.: Extended formulations for combinatorial polytopes. Ph.D. thesis, Magdeburg Universität (2012)

  17. Rothvoß, T.: The matching polytope has exponential extension complexity. In: Proceedings of STOC, pp. 263–272 (2014)

  18. Schoenebeck, G.: Linear level Lasserre lower bounds for certain k-CSPs. In: IEEE 49th Annual IEEE Symposium on Foundations of Computer Science FOCS’08, pp. 593–602. IEEE (2008)

  19. Trevisan, L., Sorkin, G.B., Sudan, M., Williamson, D.P.: Gadgets, approximation, and linear programming. SIAM J. Comput. 29(6), 2074–2097 (2000)

    Article  MathSciNet  Google Scholar 

  20. Yannakakis, M.: Expressing combinatorial optimization problems by linear programs (extended abstract). In: Proceedings of STOC, pp. 223–228 (1988)

  21. Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991). https://doi.org/10.1016/0022-0000(91)90024-Y

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Parts of this research was conducted at the CMO-BIRS 2015 workshop Modern Techniques in Discrete Optimization: Mathematics, Algorithms and Applications and we would like to thank the organizers for providing a stimulating research environment.

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Authors and Affiliations

  1. ISyE, Georgia Institute of Technology, Atlanta, GA, USA

    Gábor Braun & Sebastian Pokutta

  2. College of Computing, Georgia Institute of Technology, Atlanta, GA, USA

    Aurko Roy

Authors
  1. Gábor Braun
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  2. Sebastian Pokutta
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  3. Aurko Roy
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Corresponding author

Correspondence to Gábor Braun.

Additional information

Research reported in this paper was partially supported by NSF CAREER award CMMI-1452463.

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Braun, G., Pokutta, S. & Roy, A. Strong reductions for extended formulations. Math. Program. 172, 591–620 (2018). https://doi.org/10.1007/s10107-018-1316-y

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  • DOI: https://doi.org/10.1007/s10107-018-1316-y

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