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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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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.

Cite AsGet 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

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.

Subject Classification

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

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