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On the Approximability of Multistage Min-Sum Set Cover

Authors Dimitris Fotakis , Panagiotis Kostopanagiotis, Vasileios Nakos, Georgios Piliouras, Stratis Skoulakis

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation Algorithms
  • Multistage Min-Sum Set Cover
  • Multistage Optimization Problems

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Abstract

We investigate the polynomial-time approximability of the multistage version of Min-Sum Set Cover (Mult-MSSC), a natural and intriguing generalization of the classical List Update problem. In Mult-MSSC, we maintain a sequence of permutations (π⁰, π¹, …, π^T) on n elements, based on a sequence of requests ℛ = (R¹, …, R^T). We aim to minimize the total cost of updating π^{t-1} to π^{t}, quantified by the Kendall tau distance d_{KT}(π^{t-1}, π^t), plus the total cost of covering each request R^t with the current permutation π^t, quantified by the position of the first element of R^t in π^t. 
Using a reduction from Set Cover, we show that Mult-MSSC does not admit an O(1)-approximation, unless P = NP, and that any o(log n) (resp. o(r)) approximation to Mult-MSSC implies a sublogarithmic (resp. o(r)) approximation to Set Cover (resp. where each element appears at most r times). Our main technical contribution is to show that Mult-MSSC can be approximated in polynomial-time within a factor of O(log² n) in general instances, by randomized rounding, and within a factor of O(r²), if all requests have cardinality at most r, by deterministic rounding.

Cite As Get BibTex

Dimitris Fotakis, Panagiotis Kostopanagiotis, Vasileios Nakos, Georgios Piliouras, and Stratis Skoulakis. On the Approximability of Multistage Min-Sum Set Cover. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 65:1-65:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.65

Author Details

Dimitris Fotakis
  • National Technical University of Athens, Greece
Panagiotis Kostopanagiotis
  • National Technical University of Athens, Greece
Vasileios Nakos
  • Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Georgios Piliouras
  • Singapore University of Technology and Design, Singapore
Stratis Skoulakis
  • Singapore University of Technology and Design, Singapore

Funding

  • Fotakis, Dimitris: Supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the "First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant", project BALSAM, HFRI-FM17-1424.
  • Nakos, Vasileios: Supported by the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).
  • Piliouras, Georgios: Supported by NRF2019-NRF-ANR095 ALIAS grant, grant PIE-SGP-AI-2018-01, NRF 2018 Fellowship NRF-NRFF2018-07, AME Programmatic Fund (Grant No. A20H6b0151) from the Agency for Science, Technology and Research (A*STAR) and AI Singapore grant AISG2-RP-2020-016.
  • Skoulakis, Stratis: Supported by NRF 2018 Fellowship NRF-NRFF2018-07.

References

  1. Noga Alon, Dana Moshkovitz, and Shmuel Safra. Algorithmic construction of sets for k-restrictions. ACM Transactions on Algorithms (TALG), 2(2):153-177, 2006. Google Scholar
  2. Christoph Ambühl. Offline list update is NP-hard. In Algorithms - ESA 2000, 8th Annual European Symposium, Proceedings, volume 1879 of Lecture Notes in Computer Science, pages 42-51. Springer, 2000. Google Scholar
  3. Hyung-Chan An, Ashkan Norouzi-Fard, and Ola Svensson. Dynamic facility location via exponential clocks. ACM Trans. Algorithms, 13(2):21:1-21:20, 2017. Google Scholar
  4. Yossi Azar and Iftah Gamzu. Ranking with submodular valuations. In SODA, pages 1070-1079, 2011. URL: https://doi.org/10.1137/1.9781611973082.81.
  5. Yossi Azar, Iftah Gamzu, and Xiaoxin Yin. Multiple intents re-ranking. In STOC, pages 669-678, 2009. URL: https://doi.org/10.1145/1536414.1536505.
  6. Nikhil Bansal, Jatin Batra, Majid Farhadi, and Prasad Tetali. Improved approximations for min sum vertex cover and generalized min sum set cover. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 998-1005. SIAM, 2021. Google Scholar
  7. Nikhil Bansal, Niv Buchbinder, and Joseph Naor. A primal-dual randomized algorithm for weighted paging. J. ACM, 59(4):19:1-19:24, 2012. URL: https://doi.org/10.1145/2339123.2339126.
  8. Nikhil Bansal, Anupam Gupta, and Ravishankar Krishnaswamy. A constant factor approximation algorithm for generalized min-sum set cover. In SODA, pages 1539-1545, 2010. URL: https://doi.org/10.1137/1.9781611973075.125.
  9. Amotz Bar-Noy, Mihir Bellare, Magnús M. Halldórsson, Hadas Shachnai, and Tami Tamir. On chromatic sums and distributed resource allocation. Inf. Comput., 140(2):183-202, 1998. URL: https://doi.org/10.1006/inco.1997.2677.
  10. Omer Ben-Porat and Moshe Tennenholtz. A game-theoretic approach to recommendation systems with strategic content providers. In Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, 2018. Google Scholar
  11. Guillaume Cabanac and Thomas Preuss. Capitalizing on order effects in the bids of peer-reviewed conferences to secure reviews by expert referees. J. Am. Soc. Inf. Sci. Technol., 64(2):405–415, 2013. URL: https://doi.org/10.1002/asi.22747.
  12. Mahsa Derakhshan, Negin Golrezaei, Vahideh Manshadi, and Vahab Mirrokni. Product ranking on online platforms. In Proc. of the 21st ACM Conference on Economics and Computation (EC 2015). ACM, 2020. URL: https://ssrn.com/abstract=3130378.
  13. Cynthia Dwork, Ravi Kumar, Moni Naor, and D. Sivakumar. Rank aggregation methods for the web. In Proceedings of the 10th International Conference on World Wide Web, WWW ’01, page 613–622, New York, NY, USA, 2001. Association for Computing Machinery. Google Scholar
  14. David Eisenstat, Claire Mathieu, and Nicolas Schabanel. Facility location in evolving metrics. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings, Part II, volume 8573 of Lecture Notes in Computer Science, pages 459-470. Springer, 2014. Google Scholar
  15. Uriel Feige, László Lovász, and Prasad Tetali. Approximating min sum set cover. Algorithmica, 40(4):219-234, 2004. URL: https://doi.org/10.1007/s00453-004-1110-5.
  16. Tanner Fiez, Nihar Shah, and Lillian Ratliff. A super* algorithm to determine orderings of items to show users. In Conference on Uncertainty in Artificial Intelligence, 2020. Google Scholar
  17. Dimitris Fotakis, Loukas Kavouras, Grigorios Koumoutsos, Stratis Skoulakis, and Manolis Vardas. The online min-sum set cover problem. In Proc. of the 47th International Colloquium on Automata, Languages and Programming (ICALP 2020), LIPIcs, 2020. Google Scholar
  18. Dimitris Fotakis, Thanasis Lianeas, Georgios Piliouras, and Stratis Skoulakis. Efficient online learning of optimal rankings: Dimensionality reduction via gradient descent. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, 2020. Google Scholar
  19. Anupam Gupta, Kunal Talwar, and Udi Wieder. Changing bases: Multistage optimization for matroids and matchings. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 563-575. Springer, 2014. Google Scholar
  20. Refael Hassin and Asaf Levin. An approximation algorithm for the minimum latency set cover problem. In ESA, pages 726-733, 2005. URL: https://doi.org/10.1007/11561071_64.
  21. Sungjin Im. Min-sum set cover and its generalizations. In Encyclopedia of Algorithms, pages 1331-1334. Springer, 2016. Google Scholar
  22. Sungjin Im, Viswanath Nagarajan, and Ruben van der Zwaan. Minimum latency submodular cover. ACM Trans. Algorithms, 13(1):13:1-13:28, 2016. URL: https://doi.org/10.1145/2987751.
  23. Sungjin Im, Maxim Sviridenko, and Ruben van der Zwaan. Preemptive and non-preemptive generalized min sum set cover. Math. Program., 145(1-2):377-401, 2014. URL: https://doi.org/10.1007/s10107-013-0651-2.
  24. Alejandro López-Ortiz, Marc P. Renault, and Adi Rosén. Paid exchanges are worth the price. Theoretical Computer Science, 824-825:1-10, 2020. Google Scholar
  25. Martin Skutella and David P. Williamson. A note on the generalized min-sum set cover problem. Oper. Res. Lett., 39(6):433-436, 2011. URL: https://doi.org/10.1016/j.orl.2011.08.002.
  26. Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. J. ACM, 32(3):652-686, 1985. URL: https://doi.org/10.1145/3828.3835.
  27. Matthew J. Streeter, Daniel Golovin, and Andreas Krause. Online learning of assignments. In Advances in Neural Information Processing Systems 22: 23rd Annual Conference on Neural Information Processing Systems 2009, pages 1794-1802. Curran Associates, Inc., 2009. Google Scholar
  28. Erez Timnat. The list update problem, 2016. Master Thesis, Technion- Israel Institute of Technology. Google Scholar
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