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Maximum Generalized Assignment with Convex Costs

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Combinatorial Optimization (ISCO 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8596))

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Abstract

We consider a generalization of the maximum generalized assignment problem. We relax the hard constraints for the bin capacities, and introduce for every bin a cost function that is convex in the total load on this bin. These costs are subtracted from the profits of assigned items, and the task is to find an assignment maximizing the resulting net profit.

We show that even restricted cases of this problem remain strongly NP-complete, and identify two cases that can be solved in strongly polynomial time. Furthermore, we present a \((1-1/e)\)-approximation algorithm for the general case. This algorithm uses a configuration based integer programming formulation for a randomized rounding procedure. In order to turn the rounded solution into a feasible solution, we define appropriate estimators that linearize the convex costs.

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Notes

  1. 1.

    Observe that the function \(h(w):=w- \frac{w^2}{2B}\) attains its global maximum \(\frac{B}{2}\) for \(w=B\).

  2. 2.

    Let \(B+\delta _j\) be the load on bin \(j\), where \(\sum _{j\in \mathcal {B}} \delta _j = 0\). Then, the total costs are \(\sum _{j \in \mathcal {B}} c_j(B+\delta _j) = \frac{1}{2B} \left( mB^2 + \sum _{j\in \mathcal {B}} \delta ^2_j\right) = \frac{mB}{2},\) which implies that \(\delta _j = 0\) for all \(j\).

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Acknowledgments

This research was partially supported by the German Research Foundation (DFG), grant GRK 1703/1 for the Research Training Group “Resource Efficiency in Interorganizational Networks – Planning Methods to Utilize Renewable Resources”.

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Authors and Affiliations

  1. Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083, Göttingen, Germany

    Marco Bender & Stephan Westphal

Authors
  1. Marco Bender
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  2. Stephan Westphal
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Correspondence to Marco Bender .

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Editors and Affiliations

  1. Univ Pierre et Marie Curie LIP6, Paris, France

    Pierre Fouilhoux

  2. Universidade de Lisboa, Lisbon, Portugal

    Luis Eduardo Neves Gouveia

  3. LAMSADE, Université Paris-Dauphine, Paris Cedex 16, France

    A. Ridha Mahjoub

  4. LAMSADE, Université Paris-Dauphine, Paris Cedex 16, France

    Vangelis T. Paschos

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Bender, M., Westphal, S. (2014). Maximum Generalized Assignment with Convex Costs. In: Fouilhoux, P., Gouveia, L., Mahjoub, A., Paschos, V. (eds) Combinatorial Optimization. ISCO 2014. Lecture Notes in Computer Science(), vol 8596. Springer, Cham. https://doi.org/10.1007/978-3-319-09174-7_7

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  • DOI: https://doi.org/10.1007/978-3-319-09174-7_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-09173-0

  • Online ISBN: 978-3-319-09174-7

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