Abstract
The problem of partitioning an orthogonal polyhedron into a minimum number of boxes was shown to be NP-hard in 1991, but no approximability result is known except for a 4-approximation algorithm for 3D-histograms. In this paper we broaden the understanding of the 3D-histogram partitioning problem. We prove that partitioning a 3D-histogram into a minimum number of boxes is NP-hard, even for histograms of height two. This settles an open question posed by Floderus et al. We then show the problem to be APX-hard for histograms of height four. On the positive side, we give polynomial-time algorithms to compute optimal or approximate box partitions for some restricted but interesting classes of polyhedra and 3D-histograms.
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References
Alimonti, P., Kann, V.: Some APX-completeness results for cubic graphs. Theoret. Comput. Sci. 237(1–2), 123–134 (2000)
Barrera-Cruz, F., Biedl, T.C., Derka, M., Kiazyk, S., Lubiw, A., Vosoughpour, H.: Turning orthogonally convex polyhedra into orthoballs. In: Proceedings of CCCG (2014)
Dielissen, V.J., Kaldewaij, A.: Rectangular partition is polynomial in two dimensions but NP-complete in three. Inf. Process. Lett. 38(1), 1–6 (1991)
Durocher, S., Mehrabi, S.: Computing conforming partitions of orthogonal polygons with minimum stabbing number. Theor. Comput. Sci. 689, 157–168 (2017)
Eppstein, D.: Graph-theoretic solutions to computational geometry problems. In: Paul, C., Habib, M. (eds.) WG 2009. LNCS, vol. 5911, pp. 1–16. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11409-0_1
Eppstein, D., Mumford, E.: Steinitz theorems for simple orthogonal polyhedra. J. Comput. Geom. 5(1), 179–244 (2014)
Ferrari, L., Sankar, P.V., Sklansky, J.: Minimal rectangular partitions of digitized blobs. Comput. Vis. Graph. Image Process. 28(1), 58–71 (1984)
Floderus, P., Jansson, J., Levcopoulos, C., Lingas, A., Sledneu, D.: 3D rectangulations and geometric matrix multiplication. Algorithmica (2016, in press)
Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)
Keil, M., Snoeyink, J.: On the time bound for convex decomposition of simple polygons. Int. J. Comput. Geom. Appl. 12(03), 181–192 (2002)
Lingas, A., Pinter, R., Rivest, R., Shamir, A.: Minimum edge length partitioning of rectilinear polygons. In: Proceedings of the Annual Allerton Conference on Communication, Control, and Computing, vol. 10, pp. 53–63 (1982)
Lipski, W., Lodi, E., Luccio, F., Mugnai, C., Pagli, L.: On two-dimensional data organization II. Fundam. Informaticae 2, 245–260 (1979)
Lipski, W.: Finding a Manhattan path and related problems. Networks 13(3), 399–409 (1983)
Ohtsuki, T.: Minimum dissection of rectilinear regions. In: Proceedings of the IEEE International Symposium on Circuits and Systems, pp. 1210–1213 (1982)
O’Rourke, J., Supowit, K.J.: Some NP-hard polygon decomposition problems. IEEE Trans. Inf. Theory 29(2), 181–189 (1983)
O’Rourke, J., Tewari, G.: The structure of optimal partitions of orthogonal polygons into fat rectangles. Comput. Geom. 28(1), 49–71 (2004)
Poljak, S.: A note on stable sets and colorings of graphs. Comment. Math. Univ. Carol. 15(2), 307–309 (1974)
Uehara, R.: NP-complete problems on a 3-connected cubic planar graph and their applications. Technical report TWCU-M-0004, Tokyo Woman’s Christian University (1996)
Acknowledgements
This work was done as part of the Algorithms Problem Session at the University of Waterloo. We thank the other participants for valuable discussions. Research of T.B. and A.L. supported by NSERC, M.D. supported by Vanier CGS, M.D. and D.M. supported by an NSERC PDF.
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Biedl, T., Derka, M., Irvine, V., Lubiw, A., Mondal, D., Turcotte, A. (2018). Partitioning Orthogonal Histograms into Rectangular Boxes. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_12
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