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A PTAS for the Time-Invariant Incremental Knapsack Problem

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Combinatorial Optimization (ISCO 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10856))

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Abstract

The Time-Invariant Incremental Knapsack problem (IIK) is a generalization of Maximum Knapsack to a discrete multi-period setting. At each time, capacity increases and items can be added, but not removed from the knapsack. The goal is to maximize the sum of profits over all times. IIK models various applications including specific financial markets and governmental decision processes. IIK is strongly NP-Hard [2] and there has been work [2, 3, 6, 13, 15] on giving approximation algorithms for some special cases. In this paper, we settle the complexity of IIK by designing a PTAS based on rounding a disjunctive formulation, and provide several extensions of the technique.

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Notes

  1. 1.

    See the full version of the paper [4] for a discussion on disjunctive programming.

References

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Acknowledgements

We thank Andrey Kupavskii for valuable combinatorial insights on the topic. Yuri Faenza’s research was partially supported by the SNSF Ambizione fellowship PZ00P2\(\_\)154779 Tight formulations of 0 / 1 problems. Some of the work was done when Igor Malinović visited Columbia University partially funded by a gift from the SNSF.

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Authors and Affiliations

  1. Columbia University, New York City, USA

    Yuri Faenza

  2. École Polytechnique Fédéral de Lausanne, Lausanne, Switzerland

    Igor Malinovic

Authors
  1. Yuri Faenza
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  2. Igor Malinovic
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Corresponding author

Correspondence to Igor Malinovic .

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Editors and Affiliations

  1. Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI, USA

    Jon Lee

  2. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” (IASI), CNR, Rome, Italy

    Giovanni Rinaldi

  3. LAMSADE, Université Paris-Dauphine, Paris, France

    A. Ridha Mahjoub

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Faenza, Y., Malinovic, I. (2018). A PTAS for the Time-Invariant Incremental Knapsack Problem. In: Lee, J., Rinaldi, G., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2018. Lecture Notes in Computer Science(), vol 10856. Springer, Cham. https://doi.org/10.1007/978-3-319-96151-4_14

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  • DOI: https://doi.org/10.1007/978-3-319-96151-4_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-96150-7

  • Online ISBN: 978-3-319-96151-4

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