Abstract
We design a 1.75-approximation algorithm for a special case of scheduling parallel machines to minimize the makespan, namely the case where each job can be assigned to at most two machines, with the same processing time on either machine. (This is a special case of so-called restricted assignment, where the set of eligible machines can be arbitrary for each job.) This is the first improvement of the approximation ratio 2 of Lenstra, Shmoys, and Tardos (Math. Program. 46:259–271, 1990), to a smaller constant for any special case with unbounded number of machines and unbounded processing times. We conclude by showing integrality gaps of several relaxations of related problems.
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Acknowledgements
We are grateful to Yossi Azar, Nikhil Bansal and Maxim Sviridenko for discussions concerning this problem. We are grateful to Eiji Miyano for pointing out the work of their group on the same problem. We are also grateful to the anonymous referees for numerous and insightful comments that helped to improve our presentation.
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Partially supported by the Center of Excellence–Inst. for Theor. Comp. Sci., Prague (project P202/12/G061 of GA ČR and project 1M0545 of MŠMT ČR), grant IAA100190902 of GA AV ČR, and project GAUK 166610.
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Ebenlendr, T., Krčál, M. & Sgall, J. Graph Balancing: A Special Case of Scheduling Unrelated Parallel Machines. Algorithmica 68, 62–80 (2014). https://doi.org/10.1007/s00453-012-9668-9
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DOI: https://doi.org/10.1007/s00453-012-9668-9