Maximum Box Problem on Stochastic Points

  • Luis Evaristo Caraballo
  • Pablo Pérez-Lantero
  • Carlos Seara
  • Inmaculada Ventura
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


Given a finite set of weighted points in \(\mathbb {R}^d\) (where there can be negative weights), the maximum box problem asks for an axis-aligned rectangle (i.e., box) such that the sum of the weights of the points that it contains is maximized. We consider that each point of the input has a probability of being present in the final random point set, and these events are mutually independent; then, the total weight of a maximum box is a random variable. We aim to compute both the probability that this variable is at least a given parameter, and its expectation. We show that even in \(d=1\) these computations are #P-hard, and give pseudo polynomial-time algorithms in the case where the weights are integers in a bounded interval. For \(d=2\), we consider that each point is colored red or blue, where red points have weight \(+1\) and blue points weight \(-\infty \). The random variable is the maximum number of red points that can be covered with a box not containing any blue point. We prove that the above two computations are also #P-hard, and give a polynomial-time algorithm for computing the probability that there is a box containing exactly two red points, no blue point, and a given point of the plane.



L.E.C. and I.V. are supported by MTM2016-76272-R (AEI/ FEDER, UE). L.E.C. is also supported by the Spanish Government under the FPU grant agreement FPU14/04705. P.P.L. is supported by CONICYT FONDECYT/Regular 1160543 (Chile). C.S. is supported by Gen. Cat. DGR 2014SGR46 and MINECO MTM2015-63791-R.


  1. 1.
    Agarwal, P.K., Kumar, N., Sintos, S., Suri, S.: Range-max queries on uncertain data. J. Comput. Syst. Sci. (2017)Google Scholar
  2. 2.
    Alliez, P., Devillers, O., Snoeyink, J.: Removing degeneracies by perturbing the problem or perturbing the world. Reliab. Comput. 6(1), 61–79 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Backer, J., Keil, J.M.: The mono- and bichromatic empty rectangle and square problems in all dimensions. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 14–25. Springer, Heidelberg (2010). CrossRefGoogle Scholar
  4. 4.
    Barbay, J., Chan, T.M., Navarro, G., Pérez-Lantero, P.: Maximum-weight planar boxes in \({O}(n^2)\) time (and better). Inf. Process. Lett. 114(8), 437–445 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caraballo, L.E., Ochoa, C., Pérez-Lantero, P., Rojas-Ledesma, J.: Matching colored points with rectangles. J. Comb. Optim. 33(2), 403–421 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chan, T.M., Kamousi, P., Suri, S.: Stochastic minimum spanning trees in Euclidean spaces. In: SoCG 2011, pp. 65–74 (2011)Google Scholar
  7. 7.
    Cortés, C., Díaz-Báñez, J.M., Pérez-Lantero, P., Seara, C., Urrutia, J., Ventura, I.: Bichromatic separability with two boxes: a general approach. J. Algorithms 64(2–3), 79–88 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Faliszewski, P., Hemaspaandra, L.: The complexity of power-index comparison. Theor. Comput. Sci. 410(1), 101–107 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Feldman, D., Munteanu, A., Sohler, C.: Smallest enclosing ball for probabilistic data. In: SoCG 2014, pp. 214–223 (2014)Google Scholar
  10. 10.
    Fink, M., Hershberger, J., Kumar, N., Suri, S.: Hyperplane separability and convexity of probabilistic point sets. JCG 8(2), 32–57 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kamousi, P., Chan, T.M., Suri, S.: Closest pair and the post office problem for stochastic points. Comput. Geom. 47(2), 214–223 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pérez-Lantero, P.: Area and perimeter of the convex hull of stochastic points. Comput. J. 59(8), 1144–1154 (2016)CrossRefGoogle Scholar
  13. 13.
    Suri, S., Verbeek, K., Yıldız, H.: On the most likely convex hull of uncertain points. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 791–802. Springer, Heidelberg (2013). CrossRefGoogle Scholar
  14. 14.
    Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31(2), 398–427 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Trans. Comput. 100(2), 135–140 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dept. de Matemática Aplicada IIUniversidad de SevillaSevilleSpain
  2. 2.Dept. de Matemática y Ciencia de la ComputaciónUniversidad de SantiagoSantiagoChile
  3. 3.Dept. de MatemàtiquesUniversitat Politècnica de CatalunyaBarcelonaSpain

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