Efficient algorithms for k-regret minimizing sets

A regret minimizing set Q is a small size representation of a much larger database P so that user queries executed on Q return answers whose scores are not much worse than those on the full dataset. In particular, a k-regret minimizing set has the property that the regret ratio between the score of the top-1 item in Q and the score of the top-k item in P is minimized, where the score of an item is the inner product of the item's attributes with a user's weight (preference) vector. The problem is challenging because we want to find a single representative set Q whose regret ratio is small with respect to all possible user weight vectors. We show that k-regret minimization is NP-Complete for all dimensions d ≥ 3, settling an open problem from Chester et al. [VLDB 2014]. Our main algorithmic contributions are two approximation algorithms, both with provable guarantees, one based on coresets and another based on hitting sets. We perform extensive experimental evaluation of our algorithms, using both real-world and synthetic data, and compare their performance against the solution proposed in [VLDB 14]. The results show that our algorithms are significantly faster and scalable to much larger sets than the greedy algorithm of Chester et al. for comparable quality answers.