Maximum Box Problem on Stochastic Points

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.

This work is the union of two works that appear in the book of abstracts of the XVII Spanish Meeting on Computational Geometry, Alicante, Spain, 2017. The extended version is available at https://sites.google.com/a/usach.cl/pablo/maxbox.pdf.

This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734922.