Verification Problem of Maximal Points under Uncertainty

The study of algorithms that handle imprecise input data for which precise data can be requested is an interesting area. In the verification under uncertainty setting, which is the focus of this paper, an algorithm is also given an assumed set of precise input data. The aim of the algorithm is to update the smallest set of input data such that if the updated input data is the same as the corresponding assumed input data, a solution can be calculated. We study this setting for the maximal point problem in two dimensions. Here there are three types of data, a set of points P = {p1, . . . , pn}, the uncertainty areas information consisting of areas of uncertainty Ai for each 1 ≤ i ≤ n, with pi ∈ Ai, and the set of P ′ = {p′1, ..., p ′ k} containing the assumed points, with p ′ i ∈ Ai. An update of an area Ai reveals the actual location of pi and verifies the assumed location if p′i = pi. The objective of an algorithm is to compute the smallest set of points with the property that, if the updates of these points verify the assumed data, the set of maximal points of P can be computed. We show that the maximal point verification problem is NP-hard, by a reduction from the minimum set cover problem.