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The Online Min-Sum Set Cover Problem

Authors Dimitris Fotakis , Loukas Kavouras, Grigorios Koumoutsos, Stratis Skoulakis, Manolis Vardas

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Author Details

Dimitris Fotakis
  • National Technical University of Athens, Greece
Loukas Kavouras
  • National Technical University of Athens, Greece
Grigorios Koumoutsos
  • Université libre de Bruxelles, Belgium
Stratis Skoulakis
  • Singapore University of Technology and Design, Singapore
Manolis Vardas
  • ETH Zurich, Switzerland

Funding

  • Fotakis, Dimitris: Partially 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 (id: 1424).
  • Kavouras, Loukas: Partially supported by a scholarship from the State Scholarships Foundation.
  • Koumoutsos, Grigorios: Supported by Fonds de la Recherche Scientifique-FNRS Grant no MISU F 6001. Part of this work was carried out while visiting the National Technical University of Athens, supported by FNRS Mobility Grant no 35282070.
  • Skoulakis, Stratis: Partially supported by NRF 2018 Fellowship NRF-NRFF2018-07. Part of this research was carried out while the author was a PhD student at the National Technical University of Athens.
  • Vardas, Manolis: This research was carried out while the author was an undergraduate student at the National Technical University of Athens.

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Dimitris Fotakis, Loukas Kavouras, Grigorios Koumoutsos, Stratis Skoulakis, and Manolis Vardas. The Online Min-Sum Set Cover Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.51

Abstract

We consider the online Min-Sum Set Cover (MSSC), a natural and intriguing generalization of the classical list update problem. In Online MSSC, the algorithm maintains a permutation on n elements based on subsets S₁, S₂, … arriving online. The algorithm serves each set S_t upon arrival, using its current permutation π_t, incurring an access cost equal to the position of the first element of S_t in π_t. Then, the algorithm may update its permutation to π_{t+1}, incurring a moving cost equal to the Kendall tau distance of π_t to π_{t+1}. The objective is to minimize the total access and moving cost for serving the entire sequence. We consider the r-uniform version, where each S_t has cardinality r. List update is the special case where r = 1.
We obtain tight bounds on the competitive ratio of deterministic online algorithms for MSSC against a static adversary, that serves the entire sequence by a single permutation. First, we show a lower bound of (r+1)(1-r/(n+1)) on the competitive ratio. Then, we consider several natural generalizations of successful list update algorithms and show that they fail to achieve any interesting competitive guarantee. On the positive side, we obtain a O(r)-competitive deterministic algorithm using ideas from online learning and the multiplicative weight updates (MWU) algorithm.
Furthermore, we consider efficient algorithms. We propose a memoryless online algorithm, called Move-All-Equally, which is inspired by the Double Coverage algorithm for the k-server problem. We show that its competitive ratio is Ω(r²) and 2^{O(√{log n ⋅ log r})}, and conjecture that it is f(r)-competitive. We also compare Move-All-Equally against the dynamic optimal solution and obtain (almost) tight bounds by showing that it is Ω(r √n) and O(r^{3/2} √n)-competitive.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online Algorithms
  • Competitive Analysis
  • Min-Sum Set Cover

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