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A constant-factor approximation for generalized malleable scheduling under \(M ^{\natural }\)-concave processing speeds

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Abstract

In generalized malleable scheduling, jobs can be allocated and processed simultaneously on multiple machines so as to reduce the overall makespan of the schedule. The required processing time for each job is determined by the joint processing speed of the allocated machines. We study the case that processing speeds are job-dependent \(M ^{\natural }\)-concave functions and provide a constant-factor approximation for this setting, significantly expanding the realm of functions for which such an approximation is possible. Further, we explore the connection between malleable scheduling and the problem of fairly allocating items to a set of agents with distinct utility functions, devising a black-box reduction that allows to obtain resource-augmented approximation algorithms for the latter.

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Notes

  1. Note that for the classic (non-malleable) makespan minimization problem, Assignment and Scheduling are in fact identical, as processing jobs in arbitrary order on their assigned machine turns any assignment \({\textbf{S}}\) into a schedule whose makespan equals the \(L({\textbf{S}})\). This equivalence, however, does not hold for the malleable case [10].

  2. For an illustrative example of this setting, consider harvesting operations in agriculture industry [32], with the machines representing employees with different skill sets. Jobs represent operations such as harvesting a field or straw baling, the slots represent different specialized tools/machinery available for those operations (e.g., a wheat field can be harvested using one or several combine harvesters of different size, possibly aided by tractors with trailers), and the weights reflect the different skill levels of the employees for operating a particular type of machinery. A matroid constraint prevents the simultaneous use of certain machinery (e.g., for safety reasons or space constraints).

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Acknowledgements

We thank the anonymous reviewers of both this article and the preceding conference version for their helpful suggestions. Dimitris Fotakis was 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, HFRI-FM17-1424. Jannik Matuschke was supported by FWO research project G072520N “Optimization and analytics for stochastic and robust project scheduling”. Orestis Papadigenopoulos was partially supported by the NSF Institute for Machine Learning, Award number 2019844.

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Authors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    Dimitris Fotakis

  2. Research Center for Operations Management, KU Leuven, Leuven, Belgium

    Jannik Matuschke

  3. Data Science Institute, Columbia University, New York, USA

    Orestis Papadigenopoulos

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  1. Dimitris Fotakis
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  2. Jannik Matuschke
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  3. Orestis Papadigenopoulos
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Correspondence to Jannik Matuschke.

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A preliminary version of this work has appeared in the proceedings of IPCO 2022

Dedicated to our friend and co-author Orestis Papadigenopoulos, who passed away on October 26, 2023

A preliminary version of this work has appeared in the proceedings of IPCO 2022.

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Fotakis, D., Matuschke, J. & Papadigenopoulos, O. A constant-factor approximation for generalized malleable scheduling under \(M ^{\natural }\)-concave processing speeds. Math. Program. (2024). https://doi.org/10.1007/s10107-023-02054-z

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