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Shrinking Maxima, Decreasing Costs: New Online Packing and Covering Problems

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Abstract

We consider two new variants of online integer programs that are duals. In the packing problem we are given a set of items and a collection of knapsack constraints over these items that are revealed over time in an online fashion. Upon arrival of a constraint we may need to remove several items (irrevocably) so as to maintain feasibility of the solution. Hence, the set of packed items becomes smaller over time. The goal is to maximize the number, or value, of packed items. The problem originates from a buffer-overflow model in communication networks, where items represent information units broken into multiple packets. The other problem considered is online covering: there is a universe to be covered. Sets arrive online, and we must decide for each set whether we add it to the cover or give it up. The cost of a solution is the total cost of sets taken, plus a penalty for each uncovered element. The number of sets in the solution grows over time, but its cost goes down. This problem is motivated by team formation, where the universe consists of skills, and sets represent candidates we may hire. The packing problem was introduced in Emek et al. (SIAM J Comput 41(4):728–746, 2012) for the special case where the matrix is binary; in this paper we extend the solution to general matrices with non-negative integer entries. The covering problem is introduced in this paper; we present matching upper and lower bounds on its competitive ratio.

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Notes

  1. We misuse the term “set” for simplicity.

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Acknowledgments

We thank Moti Medina for going to Berkeley to represent us.

Author information

Authors and Affiliations

  1. LIAFA, CNRS, University Paris Diderot, Paris, France

    Pierre Fraigniaud & Adi Rosén

  2. ICE-TCS, School of Computer Science, Reykjavik University, 101, Reykjavík, Iceland

    Magnús M. Halldórsson

  3. School of Electrical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel

    Boaz Patt-Shamir

  4. Faculty of Engineering, Bar-Ilan University, Ramat Gan, 52900, Israel

    Dror Rawitz

Authors
  1. Pierre Fraigniaud
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  2. Magnús M. Halldórsson
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  3. Boaz Patt-Shamir
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  4. Dror Rawitz
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  5. Adi Rosén
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Corresponding author

Correspondence to Dror Rawitz.

Additional information

Boaz Patt-Shamir and Dror Rawitz were supported in part by the Israeli Ministry of Science Technology, and Space.

A preliminary version was presented at the 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2013.

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Fraigniaud, P., Halldórsson, M.M., Patt-Shamir, B. et al. Shrinking Maxima, Decreasing Costs: New Online Packing and Covering Problems. Algorithmica 74, 1205–1223 (2016). https://doi.org/10.1007/s00453-015-9995-8

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  • DOI: https://doi.org/10.1007/s00453-015-9995-8

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