Abstract
A natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time. In this paper we present increasingly stronger LP-relaxations for this problem and their implications on the approximability of the problem. First we show that the straightforward LP, extending the approach for the original problem, has an integrality gap of 3 and yields an approximation algorithm of the same factor. By applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to \(1+\phi \), where \(\phi \) is the golden ratio. Since this bound remains tight for the seemingly stronger machine configuration LP, we propose a new job configuration LP that is based on an infinite continuum of fractional assignments of each job to the machines. We prove that this LP has a finite representation and can be solved in polynomial time up to any accuracy. Interestingly, we show that our problem cannot be approximated within a factor better than \(\frac{e}{e-1}\approx 1.582\, (\hbox {unless } \mathcal {P}=\mathcal {NP})\), which is larger than the inapproximability bound of 1.5 for the original problem.
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Acknowledgments
We would like to thank two anonymous referees for their insightful remarks, which helped improving the presentation of Sects. 4 and 6. This work was partially supported by Nucleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F, by EU-IRSES Grant EUSACOU, by the DFG Priority Programme “Algorithm Engineering” (SPP 1307), by ERC Starting Grant 335288-OptApprox, by FONDECYT Project 3130407, by the Berlin Mathematical School and by the Tinbergen Institute and ABRI-VU.
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Correa, J., Marchetti-Spaccamela, A., Matuschke, J. et al. Strong LP formulations for scheduling splittable jobs on unrelated machines. Math. Program. 154, 305–328 (2015). https://doi.org/10.1007/s10107-014-0831-8
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DOI: https://doi.org/10.1007/s10107-014-0831-8