Document Open Access Logo

k-Regret Minimizing Set: Efficient Algorithms and Hardness

Authors Wei Cao, Jian Li, Haitao Wang, Kangning Wang, Ruosong Wang, Raymond Chi-Wing Wong, Wei Zhan

Thumbnail PDF


  • Filesize: 0.7 MB
  • 19 pages

Document Identifiers

Author Details

Wei Cao
Jian Li
Haitao Wang
Kangning Wang
Ruosong Wang
Raymond Chi-Wing Wong
Wei Zhan

Cite AsGet BibTex

Wei Cao, Jian Li, Haitao Wang, Kangning Wang, Ruosong Wang, Raymond Chi-Wing Wong, and Wei Zhan. k-Regret Minimizing Set: Efficient Algorithms and Hardness. In 20th International Conference on Database Theory (ICDT 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 68, pp. 11:1-11:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


We study the k-regret minimizing query (k-RMS), which is a useful operator for supporting multi-criteria decision-making. Given two integers k and r, a k-RMS returns r tuples from the database which minimize the k-regret ratio, defined as one minus the worst ratio between the k-th maximum utility score among all tuples in the database and the maximum utility score of the r tuples returned. A solution set contains only r tuples, enjoying the benefits of both top-k queries and skyline queries. Proposed in 2012, the query has been studied extensively in recent years. In this paper, we advance the theory and the practice of k-RMS in the following aspects. First, we develop efficient algorithms for k-RMS (and its decision version) when the dimensionality is 2. The running time of our algorithms outperforms those of previous ones. Second, we show that k-RMS is NP-hard even when the dimensionality is 3. This provides a complete characterization of the complexity of k-RMS, and answers an open question in previous studies. In addition, we present approximation algorithms for the problem when the dimensionality is 3 or larger.
  • multi-criteria decision-making
  • regret minimizing set
  • top-k query


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. A greedy algorithm - the interval point cover problem. URL:
  2. Pankaj K Agarwal, Sariel Har-Peled, and Kasturi R Varadarajan. Approximating extent measures of points. Journal of the ACM (JACM), 51(4):606-635, 2004. Google Scholar
  3. S Borzsony, Donald Kossmann, and Konrad Stocker. The skyline operator. In ICDE, pages 421-430, 2001. Google Scholar
  4. C.Y. Chan, HV Jagadish, K.L. Tan, A.K.H. Tung, and Z. Zhang. Finding k-dominant skylines in high dimensional space. In SIGMOD, 2006. Google Scholar
  5. Timothy M Chan. Remarks on k-level algorithms in the plane. Manuscript, Department of Computer Science, University of Waterloo, Waterloo, Canada, 1999. Google Scholar
  6. Timothy M Chan. Faster core-set constructions and data stream algorithms in fixed dimensions. In Proceedings of the 20th Annual Symposium on Computational Geometry, pages 152-159, 2004. Google Scholar
  7. K. Chen. On coresets for k-median and k-means clustering in metric and euclidean spaces and their applications. SIAM Journal on Computing, 39(3):923-947, 2009. Google Scholar
  8. Sean Chester, Alex Thomo, S Venkatesh, and Sue Whitesides. Computing k-regret minimizing sets. Proceedings of VLDB, 7(5), 2014. Google Scholar
  9. Jan Chomicki, Paolo Ciaccia, and Niccolo' Meneghetti. Skyline queries, front and back. SIGMOD Rec., 42(3):6-18, October 2013. URL:
  10. Gautam Das and Michael T Goodrich. On the complexity of approximating and illuminating three-dimensional convex polyhedra. In Algorithms and Data Structures, pages 74-85. Springer, 1995. Google Scholar
  11. A. Deshpande, L. Rademacher, S. Vempala, and G. Wang. Matrix approximation and projective clustering via volume sampling. In Proceedings of the 17th ACM-SIAM symposium on Discrete algorithm, pages 1117-1126, 2006. Google Scholar
  12. Tamal K Dey. Improved bounds on planar k-sets and k-levels. In Foundations of Computer Science, 1997. Proceedings., 38th Annual Symposium on, pages 156-161. IEEE, 1997. Google Scholar
  13. D. Feldman and M. Langberg. A unified framework for approximating and clustering data. In Proceedings of the 43rd ACM Symposium on Theory of Computing, pages 569-578, 2011. Google Scholar
  14. P. Fraternali, D. Martinenghi, and M. Tagliasacchi. Top-k bounded diversification. In SIGMOD, 2012. Google Scholar
  15. M. Goncalves and M.E. Vidal. Top-k skyline: A unified approach. In On the Move to Meaningful Internet Systems 2005: OTM 2005 Workshops, pages 790-799. Springer, 2005. Google Scholar
  16. John Hershberger. Finding the upper envelope of n line segments in o (n log n) time. Information Processing Letters, 33(4):169-174, 1989. Google Scholar
  17. Lingxiao. Huang, Jian. Li, Jeff. Phillips, and Haitao. Wang. ε-kernel coresets for stochastic points. In ESA, 2016. Google Scholar
  18. Ihab F. Ilyas, George Beskales, and Mohamed A. Soliman. A survey of top-k query processing techniques in relational database systems. ACM Comput. Surv., 40(4):11:1-11:58, October 2008. URL:
  19. Taylor Kessler Faulkner, Will Brackenbury, and Ashwin Lall. K-regret queries with nonlinear utilities. Proc. VLDB Endow., 8(13):2098-2109, September 2015. URL:
  20. J. Lee, G. You, and S. Hwang. Personalized top-k skyline queries in high-dimensional space. Information Systems, 2009. Google Scholar
  21. C.E. Leiserson, C. Stein, R. Rivest, and T.H. Cormen. Introduction to algorithms. MIT press, 2009. Google Scholar
  22. X. Lin, Y. Yuan, Q. Zhang, and Y. Zhang. Selecting stars: The k most representative skyline operator. In ICDE, 2007. Google Scholar
  23. J. Matoušek. Randomized optimal algorithm for slope selection. Information Processing Letters, 39(4):183-187, 1991. Google Scholar
  24. D. Mindolin and J. Chomicki. Discovering relative importance of skyline attributes. VLDB, 2009. Google Scholar
  25. D. Nanongkai, A. Lall, A. D. Sarma, and K. Makino. Interactive regret minimization. In SIGMOD, 2012. Google Scholar
  26. Danupon Nanongkai, Atish Das Sarma, Ashwin Lall, Richard J Lipton, and Jun Xu. Regret-minimizing representative databases. Proceedings of the VLDB Endowment, 3(1-2):1114-1124, 2010. Google Scholar
  27. P. Peng and R. C.-W. Wong. Geometry approach for k-regret query. In ICDE, 2014. Google Scholar
  28. Franco P Preparata, Michael Ian Shamos, and Franco P Preparata. Computational geometry: an introduction, volume 5. Springer-Verlag New York, 1985. Google Scholar
  29. L. Qin, J.X. Yu, and L. Chang. Diversifying top-k results. VLDB, 2012. Google Scholar
  30. M.A. Soliman, I.F. Ilyas, and K. Chen-Chuan Chang. Top-k query processing in uncertain databases. In IEEE 23rd International Conference on Data Engineering., pages 896-905. IEEE, 2007. Google Scholar
  31. Y. Tao, L. Ding, X. Lin, and J. Pei. Distance-based representative skyline. In ICDE, 2009. Google Scholar
  32. Géza Tóth. Point sets with many k-sets. Discrete &Computational Geometry, 26(2):187-194, 2001. Google Scholar
  33. T. Xia, D. Zhang, and Y. Tao. On skylining with flexible dominance relation. In Data Engineering, 2008. ICDE 2008. IEEE 24th International Conference on, pages 1397-1399. IEEE, 2008. Google Scholar
  34. M.L. Yiu and N. Mamoulis. Multi-dimensional top-k dominating queries. The VLDB Journal, 18(3):695-718, 2009. Google Scholar
  35. Hai Yu, Pankaj K Agarwal, Raghunath Poreddy, and Kasturi R Varadarajan. Practical methods for shape fitting and kinetic data structures using coresets. Algorithmica, 52(3):378-402, 2008. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail