Document Open Access Logo

Property Testing of LP-Type Problems

Authors Rogers Epstein, Sandeep Silwal

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2020.98.pdf
  • Filesize: 0.54 MB
  • 18 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • property pesting
  • LP-Type problems
  • random sampling

Metrics

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

Abstract

Given query access to a set of constraints S, we wish to quickly check if some objective function φ subject to these constraints is at most a given value k. We approach this problem using the framework of property testing where our goal is to distinguish the case φ(S) ≤ k from the case that at least an ε fraction of the constraints in S need to be removed for φ(S) ≤ k to hold. We restrict our attention to the case where (S,φ) are LP-Type problems which is a rich family of combinatorial optimization problems with an inherent geometric structure. By utilizing a simple sampling procedure which has been used previously to study these problems, we are able to create property testers for any LP-Type problem whose query complexities are independent of the number of constraints. To the best of our knowledge, this is the first work that connects the area of LP-Type problems and property testing in a systematic way. Among our results are property testers for a variety of LP-Type problems that are new and also problems that have been studied previously such as a tight upper bound on the query complexity of testing clusterability with one cluster considered by Alon, Dar, Parnas, and Ron (FOCS 2000). We also supply a corresponding tight lower bound for this problem and other LP-Type problems using geometric constructions.

Cite As Get BibTex

Rogers Epstein and Sandeep Silwal. Property Testing of LP-Type Problems. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 98:1-98:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.98

Author Details

Rogers Epstein
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Sandeep Silwal
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Silwal, Sandeep: Research supported by the NSF Graduate Research Fellowship under Grant No. 1122374.

Acknowledgements

We would like to thank Ronitt Rubinfeld, Piotr Indyk, Bertie Ancona, and Rikhav Shah for helpful feedback.

References

  1. N. Alon, S. Dar, M. Parnas, and D. Ron. Testing of clustering. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 240-250, November 2000. URL: https://doi.org/10.1109/SFCS.2000.892111.
  2. Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. The power and limitations of uniform samples in testing properties of figures. Algorithmica, 81(3):1247–1266, March 2019. URL: https://doi.org/10.1007/s00453-018-0467-9.
  3. Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Testing convexity of figures under the uniform distribution. Random Structures & Algorithms, 54(3):413-443, 2019. URL: https://doi.org/10.1002/rsa.20797.
  4. Yves Brise and Bernd Gärtner. Clarksons algorithm for violator spaces. CoRR, abs/0906.4706, 2009. URL: http://arxiv.org/abs/0906.4706.
  5. H. Chen, M. Valeriote, and Y. Yoshida. Testing assignments to constraint satisfaction problems. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 525-534, October 2016. URL: https://doi.org/10.1109/FOCS.2016.63.
  6. Kenneth L. Clarkson. Las vegas algorithms for linear and integer programming when the dimension is small. J. ACM, 42(2):488-499, March 1995. URL: https://doi.org/10.1145/201019.201036.
  7. Artur Czumaj and Christian Sohler. Property testing with geometric queries. In Friedhelm Meyer auf der Heide, editor, Algorithms - ESA 2001, pages 266-277, Berlin, Heidelberg, 2001. Springer Berlin Heidelberg. Google Scholar
  8. Artur Czumaj and Christian Sohler. Abstract combinatorial programs and efficient property testers. SIAM J. Comput., 34(3):580–615, March 2005. URL: https://doi.org/10.1137/S009753970444199X.
  9. Artur Czumaj, Christian Sohler, and Martin Ziegler. Property testing in computational geometry. In Mike S. Paterson, editor, Algorithms - ESA 2000, pages 155-166, Berlin, Heidelberg, 2000. Springer Berlin Heidelberg. Google Scholar
  10. B. Gärtner and E. Welzl. A simple sampling lemma: Analysis and applications in geometric optimization. Discrete & Computational Geometry, 25(4):569-590, April 2001. URL: https://doi.org/10.1007/s00454-001-0006-2.
  11. Bernd Gärtner, Jiří Matoušek, Leo Rüst, and Petr Škovroň. Violator spaces: Structure and algorithms. In Yossi Azar and Thomas Erlebach, editors, Algorithms - ESA 2006, pages 387-398, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg. Google Scholar
  12. Bernd Gärtner and Emo Welzl. Linear programming - randomization and abstract frameworks. In Claude Puech and Rüdiger Reischuk, editors, STACS 96, pages 667-687, Berlin, Heidelberg, 1996. Springer Berlin Heidelberg. Google Scholar
  13. Oded Goldreich. Introduction to Property Testing. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781108135252.
  14. Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, July 1998. URL: https://doi.org/10.1145/285055.285060.
  15. Oded Goldreich and Dana Ron. A sublinear bipartiteness tester for bounded degree graphs. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC '98, pages 289-298, New York, NY, USA, 1998. ACM. URL: https://doi.org/10.1145/276698.276767.
  16. Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, 32(2):302-343, 2002. URL: https://doi.org/10.1007/s00453-001-0078-7.
  17. Steve Hanneke. The optimal sample complexity of pac learning. J. Mach. Learn. Res., 17(1):1319–1333, January 2016. Google Scholar
  18. J. Matoušek, M. Sharir, and E. Welzl. A subexponential bound for linear programming. Algorithmica, 16(4):498-516, October 1996. URL: https://doi.org/10.1007/BF01940877.
  19. Krzysztof Onak. Testing properties of sets of points in metric spaces. In Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part I: Tack A: Algorithms, Automata, Complexity, and Games, pages 515-526, 2008. URL: https://doi.org/10.1007/978-3-540-70575-8_42.
  20. Luis Rademacher and Santosh Vempala. Testing geometric convexity. In Kamal Lodaya and Meena Mahajan, editors, FSTTCS 2004: Foundations of Software Technology and Theoretical Computer Science, pages 469-480, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. Google Scholar
  21. Sofya Raskhodnikova. Approximate testing of visual properties. In Sanjeev Arora, Klaus Jansen, José D. P. Rolim, and Amit Sahai, editors, Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques, pages 370-381, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg. Google Scholar
  22. Sofya Raskhodnikova. Approximate testing of visual properties. In Sanjeev Arora, Klaus Jansen, José D. P. Rolim, and Amit Sahai, editors, Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques, pages 370-381, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg. Google Scholar
  23. Dana Ron. Algorithmic and analysis techniques in property testing. Foundations and Trends® in Theoretical Computer Science, 5(2):73-205, 2010. URL: https://doi.org/10.1561/0400000029.
  24. Dana Ron and Gilad Tsur. Testing properties of sparse images. ACM Trans. Algorithms, 10(4), August 2014. URL: https://doi.org/10.1145/2635806.
  25. Raimund Seidel. Small-dimensional linear programming and convex hulls made easy. Discrete & Computational Geometry, 6(3):423-434, September 1991. URL: https://doi.org/10.1007/BF02574699.
  26. Micha Sharir and Emo Welzl. A combinatorial bound for linear programming and related problems. In Alain Finkel and Matthias Jantzen, editors, STACS 92, pages 567-579, Berlin, Heidelberg, 1992. Springer Berlin Heidelberg. Google Scholar
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact