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The Inverse Voronoi Problem in Graphs I: Hardness

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Abstract

We introduce the inverse Voronoi diagram problem in graphs: given a graph G with positive edge-lengths and a collection \({\mathbb {U}}\) of subsets of vertices of V(G), decide whether \({\mathbb {U}}\) is a Voronoi diagram in G with respect to the shortest-path metric. We show that the problem is NP-hard, even for planar graphs where all the edges have unit length. We also study the parameterized complexity of the problem and show that the problem is W[1]-hard when parameterized by the number of Voronoi cells or by the pathwidth of the graph.

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References

  1. Aloupis, G., Pérez-Rosés, H., Pineda-Villavicencio, G., Taslakian, P., Trinchet-Almaguer, D.: Fitting Voronoi diagrams to planar tesselations. In: Combinatorial Algorithms—24th International Workshop, IWOCA Lecture Notes in Computer Science, vol. 8288. Springer,, pp. 349–361 (2013). https://doi.org/10.1007/978-3-642-45278-9_30

  2. Ash, P.F., Bolker, E.D.: Recognizing Dirichlet tessellations. Geom. Dedic. 19(2), 175–206 (1985). https://doi.org/10.1007/BF00181470

    Article  MathSciNet  MATH  Google Scholar 

  3. Bandyapadhyay, S., Banik, A., Das, S., Sarkar, H.: Voronoi game on graphs. Theor. Comput. Sci. 562, 270–282 (2015). https://doi.org/10.1016/j.tcs.2014.10.003

    Article  MathSciNet  MATH  Google Scholar 

  4. Banerjee, S., Bhattacharya, B.B., Das, S., Karmakar, A., Maheshwari, A., Roy, S.: On the construction of generalized Voronoi inverse of a rectangular tessellation. Transactions on Computational Science. Lecture Notes in Computer Science, vol. 8110. Springer, pp. 22–38 (2013). https://doi.org/10.1007/978-3-642-41905-8_3

  5. Biniaz, A., Cabello, S., Carmi, P., De Carufel, J., Maheshwari, A., Mehrabi, S., Smid, M.: On the minimum consistent subset problem. CoRR arXiv:1810.09232 (2018)

  6. Bonnet, É., Cabello, S., Mohar, B., Pérez-Rosés, H.: The inverse Voronoi problem in graphs II: trees (2018). Manuscript. Its content is included in arXiv:1811.12547

  7. Cabello, S.: Subquadratic algorithms for the diameter and the sum of pairwise distances in planar graphs. ACM Trans. Algorithms (to appear). Preliminary version presented at SODA (2017). Full version available at arXiv:1702.07815

  8. Colin de Verdière, É.: Shortest cut graph of a surface with prescribed vertex set. In: Algorithms - ESA 2010, 18th Annual European Symposium, Part II. Lecture Notes in Computer Science, vol. 6347. Springer, pp. 100–111 (2010) https://doi.org/10.1007/978-3-642-15781-3_9

  9. Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015). https://doi.org/10.1007/978-3-319-21275-3

    Book  MATH  Google Scholar 

  10. Dey, T.K., Fan, F., Wang, Y.: Graph induced complex on point data. Comput. Geom. 48(8), 575–588 (2015). https://doi.org/10.1016/j.comgeo.2015.04.003

    Article  MathSciNet  MATH  Google Scholar 

  11. Diestel, R.: Graph Theory Graduate Texts in Mathematics, 4th edn, vol. 173. Springer, Berlin (2012)

    Google Scholar 

  12. Erwig, M.: The graph Voronoi diagram with applications. Networks 36(3), 156–163 (2000). https://doi.org/10.1002/1097-0037(200010)36:3<156::AID-NET2>3.0.CO;2-L

  13. Gawrychowski, P., Kaplan, H., Mozes, S., Sharir, M., Weimann, O.: Voronoi diagrams on planar graphs, and computing the diameter in deterministic \(\tilde{o}(n^{5/3})\) time. In: Proceedings of 29th ACM-SIAM Symposium on Discrete Algorithms, SODA 2018. SIAM, pp. 495–514 (2018). https://doi.org/10.1137/1.9781611975031.33. Full version available at arXiv:1704.02793

  14. Gawrychowski, P., Mozes, S., Weimann, O., Wulff-Nilsen, C.: Better tradeoffs for exact distance oracles in planar graphs. In: Proceedings of 29th ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, pp. 515–529 (2018). https://doi.org/10.1137/1.9781611975031.34

  15. Gerbner, D., Mészáros, V., Pálvölgyi, D., Pokrovskiy, A., Rote, G.: Advantage in the discrete Voronoi game. J. Graph Algorithms Appl. 18(3), 439–457 (2014). https://doi.org/10.7155/jgaa.00331

    Article  MathSciNet  MATH  Google Scholar 

  16. Gottlieb, L., Kontorovich, A., Nisnevitch, P.: Near-optimal sample compression for nearest neighbors. IEEE Trans. Inf. Theory 64(6), 4120–4128 (2018). https://doi.org/10.1109/TIT.2018.2822267

    Article  MathSciNet  MATH  Google Scholar 

  17. Hart, P.E.: The condensed nearest neighbor rule (corresp.). IEEE Trans. Inf. Theory 14(3), 515–516 (1968). https://doi.org/10.1109/TIT.1968.1054155

    Article  Google Scholar 

  18. Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001). https://doi.org/10.1006/jcss.2000.1727

    Article  MathSciNet  MATH  Google Scholar 

  19. Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001). https://doi.org/10.1006/jcss.2001.1774

    Article  MathSciNet  MATH  Google Scholar 

  20. Kannan, S., Mathieu, C., Zhou, H.: Graph reconstruction and verification. ACM Trans. Algorithms 14(4), 40:1–40:30 (2018). https://doi.org/10.1145/3199606. Preliminary version in ICALP 2015

    Article  MathSciNet  MATH  Google Scholar 

  21. Kirousis, L.M., Papadimitriou, C.H.: Interval graphs and seatching. Discrete Math. 55(2), 181–184 (1985). https://doi.org/10.1016/0012-365X(85)90046-9

    Article  MathSciNet  MATH  Google Scholar 

  22. Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982). https://doi.org/10.1137/0211025

    Article  MathSciNet  MATH  Google Scholar 

  23. Marx, D.: Can you beat treewidth? Theory Comput. 6(1), 85–112 (2010). https://doi.org/10.4086/toc.2010.v006a005

    Article  MathSciNet  MATH  Google Scholar 

  24. Marx, D., Pilipczuk, M.: Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams. CoRR arXiv:1504.05476 (2015). Preliminary version at ESA 2015

  25. Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM 55(2), 11:1–11:29 (2008). https://doi.org/10.1145/1346330.1346336

    Article  MathSciNet  MATH  Google Scholar 

  26. Okabe, A., Sugihara, K.: Spatial Analysis along Networks. Statistical and Computational Methods. Wiley, Hoboken (2012)

    Book  Google Scholar 

  27. Ritter, G.L., Woodruff, H.B., Lowry, S.R., Isenhour, T.L.: An algorithm for a selective nearest neighbor decision rule (corresp.). IEEE Trans. Inf. Theory 21(6), 665–669 (1975)

    Article  Google Scholar 

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Acknowledgements

Part of this work was done at the 21st Korean Workshop on Computational Geometry, held in Rogla, Slovenia, in June 2018. We thank all workshop participants for their helpful comments.

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Authors and Affiliations

  1. LIP UMR5668, CNRS, ENS de Lyon, Univ Lyon, Université Claude Bernard Lyon 1, Lyon, France

    Édouard Bonnet

  2. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sergio Cabello

  3. IMFM, Ljubljana, Slovenia

    Sergio Cabello

  4. Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada

    Bojan Mohar

  5. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Tarragona, Spain

    Hebert Pérez-Rosés

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  1. Édouard Bonnet
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Correspondence to Sergio Cabello.

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Bojan Mohar: On leave from IMFM & FMF, Department of Mathematics, University of Ljubljana.

É. Bonnet: Supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). S. Cabello: Supported by the Slovenian Research Agency, program P1-0297 and projects J1-1693, J1-8130, J1-8155, J1-9109. B. Mohar: Supported in part by the NSERC Discovery Grant R611450 (Canada), by the Canada Research Chairs program, and by the Research Project J1-8130 of ARRS (Slovenia). H. Pérez-Rosés: Partially supported by Grant MTM2017-86767-R from the Spanish Ministry of Economy, Industry and Competitiveness.

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Bonnet, É., Cabello, S., Mohar, B. et al. The Inverse Voronoi Problem in Graphs I: Hardness. Algorithmica 82, 3018–3040 (2020). https://doi.org/10.1007/s00453-020-00716-4

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