Designing Fair Ranking Schemes

Items from a database are often ranked based on a combination of multiple criteria. A user may have the flexibility to accept combinations that weigh these criteria differently, within limits. On the other hand, this choice of weights can greatly affect the fairness of the produced ranking. In this paper, we develop a system that helps users choose criterion weights that lead to greater fairness. We consider ranking functions that compute the score of each item as a weighted sum of (numeric) attribute values, and then sort items on their score. Each ranking function can be expressed as a vector of weights, or as a point in a multi-dimensional space. For a broad range of fairness criteria, we show how to efficiently identify regions in this space that satisfy these criteria. Using this identification method, our system is able to tell users whether their proposed ranking function satisfies the desired fairness criteria and, if it does not, to suggest the smallest modification that does. We develop user-controllable approximation that and indexing techniques that are applied during preprocessing, and support sub-second response times during the online phase. Our extensive experiments on real datasets demonstrate that our methods are able to find solutions that satisfy fairness criteria effectively and efficiently

Probing the arrangement of hyperplanes

AbstractIn this paper we investigate the combinatorial complexity of an algorithm to determine the geometry and the topology related to an arrangement of hyperplanes in multi-dimensional Euclidean space from the “probing” on the arrangement. The “probing” by a flat means the operation from which we can obtain the intersection of the flat and the arrangement. For a finite set H of hyperplanes in Ed, we obtain the worst-case number of fixed direction line probes and that of flat probes to determine a generic line of H and H itself. We also mention the bound for the computational complexity of these algorithms based on the efficient line probing algorithm which uses the dual transform to compute a generic line of H.We also consider the problem to approximate arrangements by extending the point probing model, which have connections with computational learning theory such as learning a network of threshold functions, and introduce the vertical probing model and the level probing model. It is shown that the former is closely related to the finger probing for a polyhedron and that the latter depends on the dual graph of the arrangement.The probing for an arrangement can be used to obtain the solution for a given system of algebraic equations by decomposing the μ-resultant into linear factors. It also has interesting applications in robotics such as a motion planning using an ultrasonic device that can detect the distances to obstacles along a specified direction