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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)

Abstract

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.

Notes

Acknowledgements

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.

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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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