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The Complexity of Recognizing Geometric Hypergraphs

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Graph Drawing and Network Visualization (GD 2023)

Abstract

As set systems, hypergraphs are omnipresent and have various representations ranging from Euler and Venn diagrams to contact representations. In a geometric representation of a hypergraph \(H=(V,E)\), each vertex \(v\in V\) is associated with a point \(p_v\in \mathbb {R}^d\) and each hyperedge \(e\in E\) is associated with a connected set \(s_e\subset \mathbb {R}^d\) such that \(\{p_v\mid v\in V\}\,\cap \, s_e=\{p_v\mid v\in e\}\) for all \(e\in E\). We say that a given hypergraph H is representable by some (infinite) family \(\mathcal {F} \) of sets in \({\mathbb {R}^d}\), if there exist \(P\subset \mathbb {R}^d\) and \(S \subseteq \mathcal {F} \) such that (PS) is a geometric representation of H. For a family \(\mathcal {F}\), we define Recognition (\(\mathcal {F}\)) as the problem to determine if a given hypergraph is representable by \(\mathcal {F}\). It is known that the Recognition problem is \(\exists \mathbb {R}\)-hard for halfspaces in \(\mathbb {R}^d\). We study the families of translates of balls and ellipsoids in \(\mathbb {R}^d\), as well as of other convex sets, and show that their Recognition problems are also \(\exists \mathbb {R}\)-complete. This means that these recognition problems are equivalent to deciding whether a multivariate system of polynomial equations with integer coefficients has a real solution.

The full version of this paper can be found on arXiv: 2302.13597v2.

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References

  1. Abrahamsen, M.: Covering polygons is even harder. In: IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS 2022), pp. 375–386 (2022). https://doi.org/10.1109/FOCS52979.2021.00045

  2. Abrahamsen, M., Adamaszek, A., Miltzow, T.: The art gallery problem is \(\exists \mathbb{R} \)-complete. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2018), pp. 65–73 (2018). https://doi.org/10.1145/3188745.3188868

  3. Abrahamsen, M., Kleist, L., Miltzow, T.: Training neural networks is \(\exists \mathbb{R}\)-complete. In: Advances in Neural Information Processing Systems (NeurIPS 2021), vol. 34, pp. 18293–18306 (2021). https://proceedings.neurips.cc/paper_files/paper/2021/file/9813b270ed0288e7c0388f0fd4ec68f5-Paper.pdf

  4. Abrahamsen, M., Kleist, L., Miltzow, T.: Geometric embeddability of complexes is \(\exists \mathbb{R} \)-complete. In: Symposium on Computational Geometry (SOCG 2023) (2023, to appear). https://doi.org/10.48550/arXiv.2108.02585

  5. Abrahamsen, M., Miltzow, T., Seiferth, N.: Framework for \(\exists \mathbb{R} \)-completeness of two-dimensional packing problems. In: IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS 2020), pp. 1014–1021 (2020). https://doi.org/10.1109/FOCS46700.2020.00098

  6. Aronov, B., Ezra, E., Sharir, M.: Small-size \(\epsilon \)-nets for axis-parallel rectangles and boxes. SIAM J. Comput. (SiComp) 39(7), 3248–3282 (2010). https://doi.org/10.1137/090762968

    Article  MathSciNet  Google Scholar 

  7. Axenovich, M., Ueckerdt, T.: Density of range capturing hypergraphs. J. Comput. Geom. (JoCG) 7(1) (2016). https://doi.org/10.20382/jocg.v7i1a1

  8. Berthelsen, M.L.T., Hansen, K.A.: On the computational complexity of decision problems about multi-player nash equilibria. In: Fotakis, D., Markakis, E. (eds.) SAGT 2019. LNCS, vol. 11801, pp. 153–167. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-30473-7_11

    Chapter  Google Scholar 

  9. Bertschinger, D., Hertrich, C., Jungeblut, P., Miltzow, T., Weber, S.: Training fully connected neural networks is \(\exists \mathbb{R} \)-complete (2022). https://doi.org/10.48550/arXiv.2204.01368

  10. Bilò, V., Mavronicolas, M.: A catalog of \(\exists \mathbb{R} \)-complete problems about Nash equilibria in multi-player games. In: 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016), pp. 17:1–17:13. (LIPIcs) (2016). https://doi.org/10.4230/LIPIcs.STACS.2016.17

  11. Bilò, V., Mavronicolas, M.: \(\exists \mathbb{R} \)-complete decision problems about symmetric Nash equilibria in symmetric multi-player games. In: 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). (LIPIcs), vol. 66, pp. 13:1–13:14 (2017). https://doi.org/10.4230/LIPIcs.STACS.2017.13

  12. Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976). https://doi.org/10.1016/S0022-0000(76)80045-1

    Article  MathSciNet  Google Scholar 

  13. Čadek, M., Krčál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., Wagner, U.: Computing all maps into a sphere. J. ACM 61(3), 1–44 (2014). https://doi.org/10.1145/2597629

    Article  MathSciNet  Google Scholar 

  14. Čadek, M., Krčál, M., Matoušek, J., Vokřínek, L., Wagner, U.: Time computation of homotopy groups and Postnikov systems in fixed dimension. SIAM J. Comput. 43(5), 1728–1780 (2014). https://doi.org/10.1137/120899029

    Article  MathSciNet  Google Scholar 

  15. Čadek, M., Krčál, M., Vokřínek, L.: Algorithmic solvability of the lifting-extension problem. Discrete Comput. Geom. (DCG) 57(4), 915–965 (2017). https://doi.org/10.1007/s00454-016-9855-6

    Article  MathSciNet  Google Scholar 

  16. Cardinal, J., Felsner, S., Miltzow, T., Tompkins, C., Vogtenhuber, B.: Intersection graphs of rays and grounded segments. J. Graph Algorithms Appl. 22(2), 273–294 (2018). https://doi.org/10.7155/jgaa.00470

    Article  MathSciNet  Google Scholar 

  17. Chistikov, D., Kiefer, S., Marusic, I., Shirmohammadi, M., Worrell, J.: On restricted nonnegative matrix factorization. In: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). LIPIcs, vol. 55, pp. 103:1–103:14 (2016). https://doi.org/10.4230/LIPIcs.ICALP.2016.103

  18. Dey, T.K., Pach, J.: Extremal problems for geometric hypergraphs. Discrete Comput. Geom. 19(4), 473–484 (1998). https://doi.org/10.1007/PL00009365

    Article  MathSciNet  Google Scholar 

  19. Dobbins, M.G., Holmsen, A., Miltzow, T.: A universality theorem for nested polytopes (2019). https://doi.org/10.48550/arXiv.1908.02213

  20. Dobbins, M.G., Kleist, L., Miltzow, T., Rzążewski, P.: Completeness for the complexity class \(\forall \exists \mathbb{R} \) and area-universality. Discrete Comput. Geom. (DCG) (2022). https://doi.org/10.1007/s00454-022-00381-0

    Article  Google Scholar 

  21. Erickson, J.: Optimal curve straightening is \(\exists \mathbb{R} \)-complete (2019). https://doi.org/10.48550/arXiv.1908.09400

  22. Erickson, J., van der Hoog, I., Miltzow, T.: Smoothing the gap between NP and \(\exists \mathbb{R} \). In: IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS 2020), pp. 1022–1033 (2020). https://doi.org/10.1109/FOCS46700.2020.00099

  23. Evans, W., Rzążewski, P., Saeedi, N., Shin, C.-S., Wolff, A.: Representing graphs and hypergraphs by touching polygons in 3D. In: Archambault, D., Tóth, C.D. (eds.) GD 2019. LNCS, vol. 11904, pp. 18–32. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-35802-0_2

    Chapter  Google Scholar 

  24. Garg, J., Mehta, R., Vazirani, V.V., Yazdanbod, S.: \(\exists \mathbb{R} \)-completeness for decision versions of multi-player (symmetric) Nash equilibria. ACM Trans. Econ. Comput. 6(1), 1:1–1:23 (2018). https://doi.org/10.1145/3175494

  25. Haussler, D., Welzl, E.: \(\varepsilon \)-nets and simplex range queries. Discrete & Comput. Geom. 127–151 (1987). https://doi.org/10.1007/BF02187876

  26. Hoffmann, M., Miltzow, T., Weber, S., Wulf, L.: Recognition of unit segment graphs is \(\exists \mathbb{R} \)-complete, unpublished (2023, in preparation)

    Google Scholar 

  27. Jungeblut, P., Kleist, L., Miltzow, T.: The complexity of the Hausdorff distance. In: 38th International Symposium on Computational Geometry (SoCG 2022). LIPIcs, vol. 224, pp. 48:1–48:17 (2022). https://doi.org/10.4230/LIPIcs.SoCG.2022.48

  28. Kang, R.J., Müller, T.: Sphere and dot product representations of graphs. Discrete Comput. Geom. 47(3), 548–568 (2012). https://doi.org/10.1007/s00454-012-9394-8

    Article  MathSciNet  Google Scholar 

  29. Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Combin. Theory Ser. B 62(2), 289–315 (1994). https://doi.org/10.1006/jctb.1994.1071

    Article  MathSciNet  Google Scholar 

  30. Looges, P.J., Olariu, S.: Optimal greedy algorithms for indifference graphs. Comput. Math. Appl. 25(7), 15–25 (1993). https://doi.org/10.1016/0898-1221(93)90308-I

    Article  MathSciNet  Google Scholar 

  31. Lubiw, A., Miltzow, T., Mondal, D.: The complexity of drawing a graph in a polygonal region. In: Biedl, T., Kerren, A. (eds.) GD 2018. LNCS, vol. 11282, pp. 387–401. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-04414-5_28

    Chapter  Google Scholar 

  32. Ma, L.: Bisectors and voronoi diagrams for convex distance functions. Ph.D. thesis, FernUniversität Hagen (2000). https://ub-deposit.fernuni-hagen.de/receive/mir_mods_00000857

  33. Matoušek, J., Sedgwick, E., Tancer, M., Wagner, U.: Embeddability in the 3-sphere is decidable. J. ACM 65(1), 1–49 (2018). https://doi.org/10.1145/2582112.2582137

    Article  MathSciNet  Google Scholar 

  34. Matoušek, J., Tancer, M., Wagner, U.: Hardness of embedding simplicial complexes in \(\mathbb{R} ^d\). JEMS 13(2), 259–295 (2011). https://doi.org/10.4171/JEMS/252

    Article  MathSciNet  Google Scholar 

  35. Matoušek, J., Seidel, R., Welzl, E.: How to net a lot with little: small \(\epsilon \)-nets for disks and halfspaces. In: Sixth Annual Symposium on Computational Geometry (SoCG 1990), pp. 16–22 (1990). https://doi.org/10.1145/98524.98530

  36. McDiarmid, C., Müller, T.: Integer realizations of disk and segment graphs. J. Combin. Theory Ser. B 103(1), 114–143 (2013). https://doi.org/10.1016/j.jctb.2012.09.004

    Article  MathSciNet  Google Scholar 

  37. Mesmay, A.D., Rieck, Y., Sedgwick, E., Tancer, M.: Embeddability in \(\mathbb{R} ^3\) is NP-hard. J. ACM 67(4), 20:1–20:29 (2020). https://doi.org/10.1145/3396593

  38. Miltzow, T., Schmiermann, R.F.: On classifying continuous constraint satisfaction problems. In: IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS 2021), pp. 781–791 (2022). https://doi.org/10.1109/FOCS52979.2021.00081

  39. Mnev, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Viro, O.Y., Vershik, A.M. (eds.) Topology and Geometry—Rohlin Seminar. LNM, vol. 1346, pp. 527–543. Springer, Heidelberg (1988). https://doi.org/10.1007/BFb0082792

    Chapter  Google Scholar 

  40. Pach, J., Tardos, G.: Tight lower bounds for the size of epsilon-nets. J. Am. Math. Soc. 26(3), 645–658 (2013). https://doi.org/10.1090/S0894-0347-2012-00759-0

    Article  MathSciNet  Google Scholar 

  41. Pach, J., Woeginger, G.: Some new bounds for epsilon-nets. In: Sixth Annual Symposium on Computational Geometry (SoCG 1990), pp. 10–15 (1990). https://doi.org/10.1145/98524.98529

  42. Richter-Gebert, J., Ziegler, G.M.: Realization spaces of 4-polytopes are universal. Bull. Am. Math. Soc. 32(4), 403–412 (1995). https://doi.org/10.1090/S0273-0979-1995-00604-X

    Article  MathSciNet  Google Scholar 

  43. Schaefer, M.: Complexity of some geometric and topological problems. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 334–344. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11805-0_32

    Chapter  Google Scholar 

  44. Schaefer, M.: Realizability of graphs and linkages. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 461–482. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-0110-0_24

    Chapter  Google Scholar 

  45. Schaefer, M.: Complexity of geometric k-planarity for fixed k. J. Graph Algorithms Appl. 25(1), 29–41 (2021). https://doi.org/10.7155/jgaa.00548

    Article  MathSciNet  Google Scholar 

  46. Schaefer, M., Sedgwick, E., Štefankovič, D.: Recognizing string graphs in NP. J. Comput. Syst. Sci. 67(2), 365–380 (2003). https://doi.org/10.1016/S0022-0000(03)00045-X

    Article  MathSciNet  Google Scholar 

  47. Schaefer, M., Štefankovič, D.: Fixed points, Nash equilibria, and the existential theory of the reals. Theory Comput. Syst. 60, 172–193 (2017). https://doi.org/10.1007/s00224-015-9662-0

    Article  MathSciNet  Google Scholar 

  48. Schaefer, M., Štefankovič, D.: The complexity of tensor rank. Theory Comput. Syst. 62(5), 1161–1174 (2018). https://doi.org/10.1007/s00224-017-9800-y

    Article  MathSciNet  Google Scholar 

  49. Shitov, Y.: A universality theorem for nonnegative matrix factorizations (2016). https://doi.org/10.48550/arXiv.1606.09068

  50. Shitov, Y.: The complexity of positive semidefinite matrix factorization. SIAM J. Optim. 27(3), 1898–1909 (2017). https://doi.org/10.1137/16M1080616

    Article  MathSciNet  Google Scholar 

  51. Shor, P.W.: Stretchability of pseudolines is NP-hard. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry And Discrete Mathematics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 531–554 (1991). https://doi.org/10.1090/dimacs/004/41

  52. Skopenkov, A.: Extendability of simplicial maps is undecidable (2020). https://doi.org/10.48550/arXiv.2008.00492

  53. Smorodinsky, S.: On the chromatic number of geometric hypergraphs. SIAM J. Discrete Math. 21(3), 676–687 (2007). https://doi.org/10.1137/050642368

    Article  MathSciNet  Google Scholar 

  54. Spinrad, J.: Recognition of circle graphs. J. Algorithms 16(2), 264–282 (1994). https://doi.org/10.1006/jagm.1994.1012

    Article  MathSciNet  Google Scholar 

  55. Stade, J.: Complexity of the boundary-guarding art gallery problem (2022). https://doi.org/10.48550/arXiv.2210.12817

  56. Tanenbaum, P.J., Goodrich, M.T., Scheinerman, E.R.: Characterization and recognition of point-halfspace and related orders. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 234–245. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-58950-3_375

    Chapter  Google Scholar 

  57. Tuncel, L., Vavasis, S., Xu, J.: Computational complexity of decomposing a symmetric matrix as a sum of positive semidefinite and diagonal matrices (2022). https://doi.org/10.48550/arXiv.2209.05678

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Bertschinger, D., El Maalouly, N., Kleist, L., Miltzow, T., Weber, S. (2023). The Complexity of Recognizing Geometric Hypergraphs. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14465. Springer, Cham. https://doi.org/10.1007/978-3-031-49272-3_12

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