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On the Configuration-LP for Scheduling on Unrelated Machines

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Algorithms – ESA 2011 (ESA 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6942))

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Abstract

Closing the approximability gap between 3/2 and 2 for the minimum makespan problem on unrelated machines is one of the most important open questions in scheduling. Almost all known approximation algorithms for the problem are based on linear programs (LPs). In this paper, we identify a surprisingly simple class of instances which constitute the core difficulty for LPs: the so far hardly studied unrelated graph balancing case in which each job can be assigned to at most two machines. We prove that already for this basic setting the strongest known LP-formulation – the configuration-LP – has an integrality gap of 2, matching the best known approximation factor for the general case. This points towards an interesting direction of future research. The result is shown by a sophisticated construction of instances, based on deep insights on two key weaknesses of the configuration-LP.

For the objective of maximizing the minimum machine load in the unrelated graph balancing setting we present an elegant purely combinatorial 2-approximation algorithm with only quadratic running time. Our algorithm uses a novel preprocessing routine that estimates the optimal value as good as the configuration-LP. This improves on the computationally costly LP-based (2 + ε)-approximation algorithm by Chakrabarty et al. [6].

This work was partially supported by Berlin Mathematical School (BMS) and by the DFG Focus Program 1307 within the project “Algorithm Engineering for Real-time Scheduling and Routing”.

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Authors and Affiliations

  1. Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    José Verschae & Andreas Wiese

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  1. José Verschae
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  2. Andreas Wiese
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Editors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, University of Rome " La Sapienza", via Ariosto 25, 00185, Roma, Italy

    Camil Demetrescu

  2. School of Computer Science, Reykjavik University, Menntavegur 1, 101, Reykjavik, Iceland

    Magnús M. Halldórsson

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Verschae, J., Wiese, A. (2011). On the Configuration-LP for Scheduling on Unrelated Machines. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_45

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  • DOI: https://doi.org/10.1007/978-3-642-23719-5_45

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23718-8

  • Online ISBN: 978-3-642-23719-5

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