An Optimal Sparsification Lemma for Low-Crossing Matchings and Its Applications to Discrepancy and Approximations

Authors Mónika Csikós , Nabil H. Mustafa



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.49.pdf
  • Filesize: 1.11 MB
  • 18 pages

Document Identifiers

Author Details

Mónika Csikós
  • Université Paris Cité, IRIF, CNRS UMR 8243 and DI-ENS, Université PSL, France
Nabil H. Mustafa
  • Université Sorbonne Paris Nord, Laboratoire LIPN, CNRS 7030, France

Cite AsGet BibTex

Mónika Csikós and Nabil H. Mustafa. An Optimal Sparsification Lemma for Low-Crossing Matchings and Its Applications to Discrepancy and Approximations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 49:1-49:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.49

Abstract

Matchings with low crossing numbers were originally introduced in the late 1980s in the seminal works of Welzl [Welzl, 1988; Welzl, 1992] and Chazelle-Welzl [Chazelle and Welzl, 1989]. They have since become fundamental structures in combinatorics, computational geometry, and algorithms. In this paper, we study matchings with low crossing numbers and their relation to random samples. In particular, our main technical result states that, given a set system (X, 𝒮) with dual VC-dimension d and a parameter α ∈ (0, 1], a random set of Θ̃(n^{1+α}) edges of binom(X,2) contains a linear-sized matching with crossing number O (n^{1-α/d}). Furthermore, we show that this bound is optimal up to a logarithmic factor. By incorporating the above sampling step to existing algorithms, we obtain improved running times, by a factor of Θ̃(n), for computing matchings with low crossing numbers. This immediately implies new bounds for a number of well-studied problems, such as combinatorial discrepancy, ε-approximations and their applications. To the best of our knowledge, these are the first near-linear time algorithms for general, non-geometric set systems, for a) matchings with sub-linear crossing numbers, and b) discrepancy beating the standard deviation bound. As an immediate consequence we get fast algorithms for computing o(1/ε²)-sized ε-approximations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • low-crossing matchings
  • uniform sampling
  • discrepancy
  • approximations

Metrics

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

References

  1. P. K. Agarwal and J. Matoušek. On range searching with semialgebraic sets. Discrete & Computational Geometry, 11(4):393-418, 1994. Google Scholar
  2. P. K. Agarwal, J. Matoušek, and M. Sharir. On range searching with semialgebraic sets. II. SIAM Journal on Computing, 42(6):2039-2062, 2013. Google Scholar
  3. N. Alon, S. Moran, and A. Yehudayoff. Sign rank versus VC dimension. In COLT, 2016. Google Scholar
  4. N. Alon and J. H. Spencer. The probabilistic method. John Wiley & Sons, 2016. Google Scholar
  5. N. Bansal. Constructive algorithms for discrepancy minimization. In Proceedings of Symposium on Foundations of Computer Science, FOCS, pages 3-10. IEEE Computer Society, 2010. Google Scholar
  6. N. Bansal, D. Dadush, S. Garg, and S. Lovett. The Gram-Schmidt walk: a cure for the Banaszczyk blues. In Proceedings of the Symposium on Theory of Computing, STOC, pages 587-597, 2018. Google Scholar
  7. N. Bansal and J. H. Spencer. Deterministic discrepancy minimization. Algorithmica, 67(4):451-471, 2013. Google Scholar
  8. T. M. Chan. Optimal partition trees. Discrete Comput. Geom., 47(4):661-690, 2012. Google Scholar
  9. T. M. Chan, E. Grant, J. Könemann, and M. Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA 2012), pages 1576-1585, 2012. Google Scholar
  10. B. Chazelle. The Discrepancy Method: Randomness and Complexity. Cambridge University Press, New York, NY, USA, 2000. Google Scholar
  11. B. Chazelle and E. Welzl. Quasi-optimal range searching in spaces of finite VC-dimension. Discrete Comput. Geom., pages 467-489, 1989. Google Scholar
  12. C. Chekuri, J. Vondrák, and R. Zenklusen. Dependent randomized rounding for matroid polytopes and applications. arXiv, abs/0909.4348, 2009. Google Scholar
  13. M. Csikós and N. H. Mustafa. Escaping the Curse of Spatial Partitioning: Matchings with Low Crossing Numbers and Their Applications. In Proceedings of Symposium on Computational Geometry (SoCG 2021), volume 189, pages 28:1-28:17, 2021. Google Scholar
  14. M. Csikós and N. H. Mustafa. Optimal approximations made easy. Inf. Process. Lett., 176:106250, 2022. Google Scholar
  15. Y. Deng, Z. Song, and O. Weinstein. Discrepancy minimization in input-sparsity time. arXiv, abs/2210.12468, 2022. URL: https://doi.org/10.48550/arXiv.2210.12468.
  16. G. Ducoffe, M. Habib, and L. Viennot. Diameter computation on h-minor free graphs and graphs of bounded (distance) VC-dimension. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1905-1922, 2020. Google Scholar
  17. S. P. Fekete, M. E. Lübbecke, and H. Meijer. Minimizing the stabbing number of matchings, trees, and triangulations. Discrete & Computational Geometry, 40(4):595-621, 2008. Google Scholar
  18. P. Giannopoulos, M. Konzack, and W Mulzer. Low-crossing spanning trees: an alternative proof and experiments. In Proceedings of EuroCG, 2014. Google Scholar
  19. S. Har-Peled. Approximating spanning trees with low crossing number. arXiv, abs/0907.1131, 2009. URL: https://arxiv.org/abs/0907.1131.
  20. D. Haussler. Sphere packing numbers for subsets of the boolean n-cube with bounded Vapnik-Chervonenkis dimension. Journal of Combinatorial Theory, Series A, 69(2):217-232, 1995. Google Scholar
  21. K. G. Larsen. Fast discrepancy minimization with hereditary guarantees. In Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, pages 276-289. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH11.
  22. A. Levy, H. Ramadas, and T. Rothvoss. Deterministic discrepancy minimization via the multiplicative weight update method. In Integer Programming and Combinatorial Optimization (IPCO), pages 380-391, 2017. Google Scholar
  23. S. Lovett and R. Meka. Constructive discrepancy minimization by walking on the edges. SIAM J. Comput., 44(5):1573-1582, 2015. Google Scholar
  24. M. Matheny and J. M. Phillips. Practical low-dimensional halfspace range space sampling. In Annual European Symposium on Algorithms (ESA), volume 112, pages 62:1-62:14, 2018. Google Scholar
  25. J. Matoušek. Geometric Discrepancy: An Illustrated Guide. Springer Berlin Heidelberg, 1999. Google Scholar
  26. J. Matoušek. Lectures on discrete geometry, volume 212. Springer Science & Business Media, 2013. Google Scholar
  27. J. Matoušek, E. Welzl, and L. Wernisch. Discrepancy and approximations for bounded VC-dimension. Combinatorica, 13(4):455-466, 1993. Google Scholar
  28. N. H. Mustafa. A simple proof of the shallow packing lemma. Discret. Comput. Geom., 55(3):739-743, 2016. Google Scholar
  29. N. H. Mustafa. Sampling in Combinatorial and Geometric Set Systems. American Mathematical Society (AMS), 2022. Google Scholar
  30. T. Rothvoss. Constructive discrepancy minimization for convex sets. SIAM J. Comput., 46(1):224-234, 2017. Google Scholar
  31. J. H. Spencer. Six standard deviations suffice. Transactions of the American Mathematical Society, 289:679-706, 1985. Google Scholar
  32. M. Talagrand. Sharper bounds for Gaussian and empirical processes. Annals of Probability, 22:28-76, 1994. Google Scholar
  33. P. Turán. On an external problem in graph theory. Mat. Fiz. Lapok, 48:436-452, 1941. Google Scholar
  34. K. R. Varadarajan. Weighted geometric set cover via quasi-uniform sampling. In Proceedings of ACM Symposium on Theory of Computing (STOC 2010), pages 641-648, 2010. Google Scholar
  35. E. Welzl. Partition trees for triangle counting and other range searching problems. In Proceedings of Annual Symposium on Computational Geometry (SoCG 1988), pages 23-33, 1988. Google Scholar
  36. E. Welzl. On spanning trees with low crossing numbers. In Data Structures and Efficient Algorithms, Final Report on the DFG Special Joint Initiative, pages 233-249, 1992. Google Scholar