A unified optimization algorithm for solving "regret-minimizing representative" problems

Abstract

Given a database with numeric attributes, it is often of interest to rank the tuples according to linear scoring functions. For a scoring function and a subset of tuples, the regret of the subset is defined as the (relative) difference in scores between the top-1 tuple of the subset and the top-1 tuple of the entire database. Finding the regret-ratio minimizing set (RRMS), i.e., the subset of a required size k that minimizes the maximum regret-ratio across all possible ranking functions, has been a well-studied problem in recent years. This problem is known to be NP-complete and there are several approximation algorithms for it. Other NP-complete variants have also been investigated, e.g., finding the set of size k that minimizes the average regret ratio over all linear functions. Prior work have designed customized algorithms for different variants of the problem, and are unlikely to easily generalize to other variants. In this paper we take a different path towards tackling these problems. In contrast to the prior, we propose a unified algorithm for solving different problem variants. Unification is done by localizing the customization to the design of variant-specific subroutines or "oracles" that are called by our algorithm. Our unified algorithm takes inspiration from the seemingly unrelated problem of clustering from data mining, and the corresponding k-medoid algorithm. We make several innovative contributions in designing our algorithm, including various techniques such as linear programming, edge sampling in graphs, volume estimation of multi-dimensional convex polytopes, and several others. We provide rigorous theoretical analysis, as well as substantial experimental evaluations over real and synthetic data sets to demonstrate the practical feasibility of our approach.