On the Complexity of Top-k Querying for Incomplete Data under Order Constraints

Obtaining data values is often expensive, e.g., when they are retrieved from human workers, from remote data repositories, or via complex computations. In this paper, we consider a scenario where the values are taken from a numerical domain, and where some partial order constraints are given over missing and known values. We study the evaluation of top-k selection queries over incomplete data in this setting. Our work is the first to provide a way to estimate the value of the top-k items (rather than just determining the items), doing so through a novel interpolation scheme, and to formally study the computational complexity of the problem. We first formulate the top-k selection problem under possible world semantics. We then present a general solution, whose complexity is polynomial in the number of possible orderings of the data tuples. While this number is unfeasibly large in the worst case, we prove that the problem is hard and thus the dependency is probably unavoidable. We then consider the case of tree-shaped partial orders, and show a polynomial-time, constructive solution.