Maximum box problem on stochastic points

Abstract

Given a finite set of weighted points in ℝd (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 - ∞. 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.