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Constructing Planar Support for Non-Piercing Regions

  • Ricky Pollack Memorial Issue
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Abstract

Given a hypergraph \(\mathcal {H}=(X,{\mathcal {S}})\), a planar support for \(\mathcal {H}\) is a planar graph G with vertex set X, such that for each hyperedge \(S\in \mathcal {S}\), the subgraph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph visualization. The main result proved in this paper is the following: given two families of regions R and B in the plane, each of which consists of connected, non-piercing regions, the intersection hypergraph \({\mathcal {H}}_R(B)=(B,\{B_{r}\}_{r\in R})\), where \(B_r=\{b\in B:b\cap r\ne \emptyset \}\) has a planar support. Further, such a planar support can be computed in time polynomial in |R|, |B|, and the number of vertices in the arrangement of the regions in \(R\cup B\). Special cases of this result include the setting where either the family R, or the family B is a set of points. Our result unifies and generalizes several previous results on planar supports, PTAS’s for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions.

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Notes

  1. A region is simply connected if any loop can be continuously shrunk to a point while staying within the region, which is true for a disk, but not for an annulus.

  2. That is, no more than two intersect at any point in the plane.

  3. An intersection graph on a set of regions is a graph whose vertices are the regions, and two vertices are adjacent if their corresponding regions intersect.

  4. See [6] for the definition of shallow cell complexity.

  5. Even though the title of the paper mentions only pseudodisks, the results in the paper apply to simply connected non-piercing regions.

  6. The notion of cell bypassing is similar in spirit to the notion of lens bypassing from [3] and to the notion of core decomposition from [20]. However, there are some technical differences being critical for the applications in this paper.

  7. The cell adjacency graph is the geometric dual of the arrangement graph in which the intersection points of the boundaries of the regions are the vertices and two vertices are adjacent in the graph if they appear consecutively along the boundary of some region.

  8. In other words H is a hole of \(\gamma \).

  9. Recall that \(\gamma \) does not contribute to the outer boundary of C.

  10. While there are infinitely many points, the hypergraph is still finite, since all points in the same cell of the arrangement \({\mathcal {A}}\) define the same hyperedge.

References

  1. Antunes, D., Mathieu, C., Mustafa, N.H.: Combinatorics of local search: an optimal \(4\)-local Hall’s theorem for planar graphs. In: 25th European Symposium on Algorithms (Vienna 2017). Leibniz Int. Proc. Inform., vol. 87, # 8. Leibniz-Zent. Inform., Wadern (2017)

  2. Aschner, R., Katz, M.J., Morgenstern, G., Yuditsky, Y.: Approximation schemes for covering and packing. In: WALCOM: Algorithms and Computation. Lecture Notes in Comput. Sci., vol. 7748, pp. 89–100. Springer, Heidelberg (2013)

  3. Basu, R.A., Govindarajan, S., Raman, R., Ray, S.: Packing and covering with non-piercing regions. Discrete Comput. Geom. 60(2), 471–492 (2018)

    Article  MathSciNet  Google Scholar 

  4. Cabello, S., Gajser, D.: Simple PTAS’s for families of graphs excluding a minor. Discrete Appl. Math. 189, 41–48 (2015)

    Article  MathSciNet  Google Scholar 

  5. Chan, T.M., Grant, E.: Exact algorithms and APX-hardness results for geometric packing and covering problems. Comput. Geom. 47(2A), 112–124 (2014)

    Article  MathSciNet  Google Scholar 

  6. Chan, T.M., Grant, E., Könemann, J., Sharpe, M.: Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (Kyoto 2012), pp. 1576–1585. ACM, New York (2012)

  7. Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. Discrete Comput. Geom. 48(2), 373–392 (2012)

    Article  MathSciNet  Google Scholar 

  8. Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry II. Discrete Comput. Geom. 4(5), 387–421 (1989)

    Article  MathSciNet  Google Scholar 

  9. Cohen-Addad, V., Klein, P.N., Mathieu, C.: Local search yields approximation schemes for \(k\)-means and \(k\)-median in Euclidean and minor-free metrics. In: 57th Annual IEEE Symposium on Foundations of Computer Science (New Brunswick 2016), pp. 353–364. IEEE Computer Soc., Los Alamitos (2016)

  10. Cohen-Addad, V., Mathieu, C.: Effectiveness of local search for geometric optimization. In: 34th International Symposium on Computational Geometry. Leibniz Int. Proc. Inform., vol. 34, pp. 329–344. Leibniz-Zent. Inform., Wadern (2015)

  11. Ene, A., Har-Peled, S., Raichel, B.: Geometric packing under non-uniform constraints. Proceedings of the 28th Annual Symposium on Computational Geometry (Chapel Hill 2012), pp. 11–20. ACM, New York (2012)

  12. Friggstad, Z., Khodamoradi, K., Rezapour, M., Salavatipour, M.R.: Approximation schemes for clustering with outliers. In: 29th Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans 2018), pp. 398–414. SIAM, Philadelphia (2018)

  13. Friggstad, Z., Rezapour, M., Salavatipour, M.R.: Local search yields a PTAS for \(k\)-means in doubling metrics. In: 57th Annual IEEE Symposium on Foundations of Computer Science (New Brunswick 2016), pp. 365–374. IEEE Computer Soc., Los Alamitos (2016)

  14. Gibson, M., Pirwani, I.A.: Algorithms for dominating set in disk graphs: breaking the \(\log n\) barrier. In: AlgorithmsESA 2010, Part I. Lecture Notes in Comput. Sci., vol. 6346, pp. 243–254. Springer, Berlin (2010)

  15. Har-Peled, S., Quanrud, K.: Approximation algorithms for polynomial-expansion and low-density graphs. SIAM J. Comput. 46(6), 1712–1744 (2017)

    Article  MathSciNet  Google Scholar 

  16. Jartoux, B., Mustafa, N.H.: Optimality of geometric local search. In: 34th International Symposium on Computational Geometry. Leibniz Int. Proc. Inform., vol. 99, # 48. Leibniz-Zent. Inform., Wadern (2018)

  17. Keller, C., Smorodinsky, S.: Conflict-free coloring of intersection graphs of geometric objects. In: 29th Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans 2018), pp. 2397–2411. SIAM, Philadelphia (2018)

  18. Keszegh, B.: Coloring intersection hypergraphs of pseudo-disks. In: 34th International Symposium on Computational Geometry. Leibniz Int. Proc. Inform., vol. 99, # 52. Leibniz-Zent. Inform., Wadern (2018)

  19. Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (2001)

    MATH  Google Scholar 

  20. Mustafa, N.H., Raman, R., Ray, S.: Quasi-polynomial time approximation scheme for weighted geometric set cover on pseudodisks and halfspaces. SIAM J. Comput. 44(6), 1650–1669 (2015)

    Article  MathSciNet  Google Scholar 

  21. Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44(4), 883–895 (2010)

    Article  MathSciNet  Google Scholar 

  22. Pyrga, E., Ray, S.: New existence proofs for \(\epsilon \)-nets. In: 24th Annual Symposium on Computational Geometry (College Park 2008), pp. 199–207. ACM, New York (2008)

  23. Raman, R., Ray, S.: Planar support for non-piercing regions and applications. In: 26th Annual European Symposium on Algorithms. Leibniz Int. Proc. Inform., vol. 112, # 69. Leibniz-Zent. Inform., Wadern (2018)

  24. Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and their Geometric Applications. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  25. Varadarajan, K.: Weighted geometric set cover via quasi-uniform sampling. In: 42nd ACM International Symposium on Theory of Computing (Cambridge 2010), pp. 641–647. ACM, New York (2010)

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Acknowledgements

The authors are indebted to the anonymous reviewers for insightful comments and corrections. One of the reviewers found a critical mistake in the time complexity analysis of cell bypassing which led to the new proof we have presented here.

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Authors and Affiliations

  1. Department of Computer Science, IIIT, Delhi, India

    Rajiv Raman

  2. Computer Science, New York University, Abu Dhabi, United Arab Emirates

    Saurabh Ray

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  1. Rajiv Raman
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  2. Saurabh Ray
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Correspondence to Rajiv Raman.

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Dedicated to the memory of Ricky Pollack

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A preliminary version has appeared in ESA 2018 [23]

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Raman, R., Ray, S. Constructing Planar Support for Non-Piercing Regions. Discrete Comput Geom 64, 1098–1122 (2020). https://doi.org/10.1007/s00454-020-00216-w

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