Document Open Access Logo

Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

Authors Per Austrin , Aleksa Stanković

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2019.24.pdf
  • Filesize: 0.52 MB
  • 17 pages

Document Identifiers

Author Details

Per Austrin
  • KTH Royal Institute of Technology, Stockholm, Sweden
Aleksa Stanković
  • KTH Royal Institute of Technology, Stockholm, Sweden

Funding

Research supported by the Approximability and Proof Complexity project funded by the Knut and Alice Wallenberg Foundation.

Acknowledgements

The authors thank Johan Håstad for helpful suggestions and comments on the manuscript. We also thank anonymous reviewers for their helpful remarks.

Cite As Get BibTex

Per Austrin and Aleksa Stanković. Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.24

Abstract

Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat).
The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Constraint satisfaction problems
  • global cardinality constraints
  • semidefinite programming
  • inapproximability
  • Unique Games Conjecture
  • Max-Cut
  • Max-2-Sat

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof Verification and Hardness of Approximation Problems. In 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, Pennsylvania, USA, 24-27 October 1992, pages 14-23, 1992. URL: https://doi.org/10.1109/SFCS.1992.267823.
  2. Sanjeev Arora and Shmuel Safra. Probabilistic Checking of Proofs; A New Characterization of NP. In 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, Pennsylvania, USA, 24-27 October 1992, pages 2-13, 1992. URL: https://doi.org/10.1109/SFCS.1992.267824.
  3. Per Austrin. Balanced max 2-sat might not be the hardest. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 189-197, 2007. URL: https://doi.org/10.1145/1250790.1250818.
  4. Per Austrin, Siavosh Benabbas, and Konstantinos Georgiou. Better Balance by Being Biased: A 0.8776-Approximation for Max Bisection. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 277-294, 2013. URL: https://doi.org/10.1137/1.9781611973105.21.
  5. Per Austrin, Subhash Khot, and Muli Safra. Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs. Theory of Computing, 7(1):27-43, 2011. URL: https://doi.org/10.4086/toc.2011.v007a003.
  6. Markus Bläser and Bodo Manthey. Improved Approximation Algorithms for Max-2SAT with Cardinality Constraint. In Algorithms and Computation, 13th International Symposium, ISAAC 2002 Vancouver, BC, Canada, November 21-23, 2002, Proceedings, pages 187-198, 2002. URL: https://doi.org/10.1007/3-540-36136-7_17.
  7. Joshua Brakensiek, Sivakanth Gopi, and Venkatesan Guruswami. CSPs with Global Modular Constraints: Algorithms and Hardness via Polynomial Representations. Electronic Colloquium on Computational Complexity (ECCC), 26:13, 2019. URL: https://eccc.weizmann.ac.il/report/2019/013.
  8. Andrei A. Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 319-330, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  9. Andrei A. Bulatov and Dániel Marx. The complexity of global cardinality constraints. Logical Methods in Computer Science, 6(4), 2010. URL: https://doi.org/10.2168/LMCS-6(4:4)2010.
  10. Irit Dinur, Elchanan Mossel, and Oded Regev. Conditional Hardness for Approximate Coloring. SIAM J. Comput., 39(3):843-873, 2009. URL: https://doi.org/10.1137/07068062X.
  11. Uriel Feige and Michael Langberg. The RPR^2 rounding technique for semidefinite programs. J. Algorithms, 60(1):1-23, 2006. URL: https://doi.org/10.1016/j.jalgor.2004.11.003.
  12. Alan M. Frieze and Mark Jerrum. Improved Approximation Algorithms for MAX k-CUT and MAX BISECTION. Algorithmica, 18(1):67-81, 1997. URL: https://doi.org/10.1007/BF02523688.
  13. Michel X. Goemans and David P. Williamson. .879-approximation algorithms for MAX CUT and MAX 2SAT. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, 23-25 May 1994, Montréal, Québec, Canada, pages 422-431, 1994. URL: https://doi.org/10.1145/195058.195216.
  14. Venkatesan Guruswami and Euiwoong Lee. Complexity of Approximating CSP with Balance / Hard Constraints. Theory Comput. Syst., 59(1):76-98, 2016. URL: https://doi.org/10.1007/s00224-015-9638-0.
  15. Venkatesan Guruswami, Rajsekar Manokaran, and Prasad Raghavendra. Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 573-582, 2008. URL: https://doi.org/10.1109/FOCS.2008.51.
  16. Eran Halperin and Uri Zwick. A unified framework for obtaining improved approximation algorithms for maximum graph bisection problems. Random Struct. Algorithms, 20(3):382-402, 2002. URL: https://doi.org/10.1002/rsa.10035.
  17. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
  18. Thomas Hofmeister. An Approximation Algorithm for MAX-2-SAT with Cardinality Constraint. In Algorithms - ESA 2003, 11th Annual European Symposium, Budapest, Hungary, September 16-19, 2003, Proceedings, pages 301-312, 2003. URL: https://doi.org/10.1007/978-3-540-39658-1_29.
  19. Howard J. Karloff and Uri Zwick. A 7/8-Approximation Algorithm for MAX 3SAT? In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, pages 406-415, 1997. URL: https://doi.org/10.1109/SFCS.1997.646129.
  20. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 767-775, 2002. URL: https://doi.org/10.1145/509907.510017.
  21. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs? SIAM J. Comput., 37(1):319-357, 2007. URL: https://doi.org/10.1137/S0097539705447372.
  22. Subhash Khot and Oded Regev. Vertex Cover Might be Hard to Approximate to within 2-ε. In 18th Annual IEEE Conference on Computational Complexity (Complexity 2003), 7-10 July 2003, Aarhus, Denmark, page 379, 2003. URL: https://doi.org/10.1109/CCC.2003.1214437.
  23. Michael Lewin, Dror Livnat, and Uri Zwick. Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems. In Integer Programming and Combinatorial Optimization, 9th International IPCO Conference, Cambridge, MA, USA, May 27-29, 2002, Proceedings, pages 67-82, 2002. URL: https://doi.org/10.1007/3-540-47867-1_6.
  24. Pasin Manurangsi. Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 954-961, 2017. URL: https://doi.org/10.1145/3055399.3055412.
  25. Pasin Manurangsi. A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation. In 2nd Symposium on Simplicity in Algorithms, SOSA@SODA 2019, January 8-9, 2019 - San Diego, CA, USA, pages 15:1-15:21, 2019. URL: https://doi.org/10.4230/OASIcs.SOSA.2019.15.
  26. Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. Ann. of Math. (2), 171(1):295-341, 2010. URL: https://doi.org/10.4007/annals.2010.171.295.
  27. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. URL: http://www.cambridge.org/de/academic/subjects/computer-science/algorithmics-complexity-computer-algebra-and-computational-g/analysis-boolean-functions.
  28. Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP? In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 245-254, 2008. URL: https://doi.org/10.1145/1374376.1374414.
  29. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 755-764. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806792.
  30. Prasad Raghavendra and Ning Tan. Approximating CSPs with global cardinality constraints using SDP hierarchies. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 373-387, 2012. URL: http://portal.acm.org/citation.cfm?id=2095149&CFID=63838676&CFTOKEN=79617016, URL: https://doi.org/10.1137/1.9781611973099.33.
  31. Ran Raz. A Parallel Repetition Theorem. SIAM J. Comput., 27(3):763-803, 1998. URL: https://doi.org/10.1137/S0097539795280895.
  32. Thomas J. Schaefer. The Complexity of Satisfiability Problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC '78, pages 216-226, New York, NY, USA, 1978. ACM. URL: https://doi.org/10.1145/800133.804350.
  33. Maxim Sviridenko. Best Possible Approximation Algorithm for MAX SAT with Cardinality Constraint. Algorithmica, 30(3):398-405, 2001. URL: https://doi.org/10.1007/s00453-001-0019-5.
  34. Yinyu Ye. A .699-approximation algorithm for Max-Bisection. Math. Program., 90(1):101-111, 2001. URL: https://doi.org/10.1007/PL00011415.
  35. Dmitriy Zhuk. A Proof of CSP Dichotomy Conjecture. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331-342, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact