Skip to main content
Springer Nature Link
Log in

Violator Spaces: Structure and Algorithms

  • Conference paper
Algorithms – ESA 2006 (ESA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4168))

Included in the following conference series:

  • 3181 Accesses

Abstract

Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson’s randomized algorithms for low-dimensional linear programming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the P-matrix generalized linear complementarity problem with a constant number of blocks. We also give two new characterizations of LP-type problems: they are equivalent to acyclic violator spaces, as well as to concrete LP-type problems (informally, the constraints in a concrete LP-type problem are subsets of a linearly ordered ground set, and the value of a set of constraints is the minimum of its intersection).

The first and the third author acknowledge support from the Swiss Science Foundation (SNF), Project No. 200020-112068/1. The fourth author acknowledges support from the Czech Science Foundation (GACR), Grant No. 201/05/H014. For a full paper see [1].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Gärtner, B., Matoušek, J., Rüst, L., Škovroň, P.: Violator spaces: Structure and algorithms. arXiv.org e-Print archive (2006)

    Google Scholar 

  2. Sharir, M., Welzl, E.: A combinatorial bound for linear programming and related problems. In: Finkel, A., Jantzen, M. (eds.) STACS 1992. LNCS, vol. 577, pp. 569–579. Springer, Heidelberg (1992)

    Google Scholar 

  3. Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16, 498–516 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kalai, G.: A subexponential randomized simplex algorithm. In: Proc. 24th Annual ACM Symposium on Theory of Computing (STOC), pp. 475–482 (1992)

    Google Scholar 

  5. Amenta, N.: Bounded boxes, Hausdorff distance, and a new proof of an interesting Helly-type theorem. In: Proc. 10th Annual Symposium on Computational Geometry (SCG), pp. 340–347. ACM Press, New York (1994)

    Chapter  Google Scholar 

  6. Amenta, N.: Helly theorems and generalized linear programming. Discrete and Computational Geometry 12, 241–261 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  7. Björklund, H., Sandberg, S., Vorobyov, S.: A discrete subexponential algorithm for parity games. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 663–674. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  8. Halman, N.: Discrete and Lexicographic Helly Theorems and Their Relations to LP-type problems. PhD thesis, Tel-Aviv University (2004)

    Google Scholar 

  9. Clarkson, K.L.: Las Vegas algorithms for linear and integer programming. Journal of the ACM 42, 488–499 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gärtner, B., Welzl, E.: Linear programming - randomization and abstract frameworks. In: Puech, C., Reischuk, R. (eds.) STACS 1996. LNCS, vol. 1046, pp. 669–687. Springer, Heidelberg (1996)

    Google Scholar 

  11. Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. Journal of Algorithms 21, 579–597 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Matoušek, J.: On geometric optimization with few violated constraints. Discrete and Computational Geometry 14, 365–384 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gärtner, B., Welzl, E.: A simple sampling lemma - analysis and applications in geometric optimization. Discrete and Computational Geometry 25(4), 569–590 (2001)

    MATH  MathSciNet  Google Scholar 

  14. Chan, T.: An optimal randomized algorithm for maximum Tukey depth. In: Proc. 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 423–429 (2004)

    Google Scholar 

  15. Amenta, N.: A short proof of an interesting Helly-type theorem. Discrete and Computational Geometry 15, 423–427 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Szabó, T., Welzl, E.: Unique sink orientations of cubes. In: Proc. 42nd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 547–555 (2000)

    Google Scholar 

  17. Gärtner, B., Morris Jr., W.D., Rüst, L.: Unique sink orientations of grids. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005. LNCS, vol. 3509, pp. 210–224. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  18. Morris Jr., W.D.: Randomized principal pivot algorithms for P-matrix linear complementarity problems. Mathematical Programming, Series A 92, 285–296 (2002)

    Article  MATH  Google Scholar 

  19. Matoušek, J.: The number of unique sink orientations of the hypercube. Combinatorica (to appear, 2006)

    Google Scholar 

  20. Morris Jr., W.D.: Distinguishing cube orientations arising from linear programs (manuscript, 2002)

    Google Scholar 

  21. Develin, M.: LP-orientations of cubes and crosspolytopes. Advances in Geometry 4, 459–468 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Matoušek, J., Szabó, T.: Random Edge can be exponential on abstract cubes. In: Proc. 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 92–100 (2004)

    Google Scholar 

  23. Schurr, I., Szabó, T.: Finding the sink takes some time. Discrete and Computational Geometry 31, 627–642 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Schurr, I., Szabó, T.: Jumping doesn’t help in abstract cubes. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005. LNCS, vol. 3509, pp. 225–235. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  25. Gärtner, B., Schurr, I.: Linear programming and unique sink orientations. In: Proc. 17th Annual Symposium on Discrete Algorithms (SODA), pp. 749–757 (2006)

    Google Scholar 

  26. Björklund, H., Vorobyov, S.: Combinatorial structure and randomized subexponential algorithms for infinite games. Theoretical Computer Science (in press, 2005)

    Google Scholar 

  27. Björklund, H., Sandberg, S., Vorobyov, S.: A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 673–685. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  28. Gärtner, B., Rüst, L.: Simple stochastic games and P-matrix generalized linear complementarity problems. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 209–220. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  29. Megiddo, N.: A note on the complexity of P-matrix LCP and computing an equilibrium. Technical report, IBM Almaden Research Center, San Jose (1988)

    Google Scholar 

  30. Škovroň, P.: Generalized linear programming. Master’s thesis, Charles University, Prague, Faculty of Mathematics and Physics (2002)

    Google Scholar 

  31. Cottle, R.W., Dantzig, G.B.: A generalization of the linear complementarity problem. Journal on Combinatorial Theory 8, 79–90 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  32. Cottle, R.W., Pang, J., Stone, R.E.: The Linear Complementarity Problem. Academic Press, London (1992)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zurich, 8092, Zurich, Switzerland

    Bernd Gärtner & Leo Rüst

  2. Department of Applied Mathematics and Institute of Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Matoušek & Petr Škovroň

Authors
  1. Bernd Gärtner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiří Matoušek
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leo Rüst
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Petr Škovroň
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA and School of Computer Science, Tel-Aviv University, 69978, Tel-Aviv, Israel

    Yossi Azar

  2. Department of Computer Science, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Thomas Erlebach

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gärtner, B., Matoušek, J., Rüst, L., Škovroň, P. (2006). Violator Spaces: Structure and Algorithms. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_36

Download citation

  • DOI: https://doi.org/10.1007/11841036_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38875-3

  • Online ISBN: 978-3-540-38876-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics