Abstract
Given a set \(L = \{J_1,J_2,\ldots ,J_n\}\) of n tasks and a set \(M = \{M_1,M_2, \ldots ,M_m\}\) of m identical machines, in which tasks and machines are possessed by different selfish clients. Each selfish client of machine \(M_i \in M\) gets a profit equal to its load and each selfish client of task allocated to \(M_i\) suffers from a cost equal to the load of \(M_i\). Our aim is to allocate the tasks on the m machines so as to minimize the maximum completion times of the tasks on each machine. A stable allocation is referred to as a dual equilibrium (DE). We firstly show that 4/3 is tight upper bound of the Price of Anarchy(PoA) with respect to dual equilibrium for \(m\in \{3,\ldots ,9\}\). And secondly \((7m-6)/(5m-3)\) is an upper bound for \(m\ge 10\). The result is better than the existing bound of 7/5.
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Acknowledgements
The authors would like to express their thanks to the National Natural Science Foundation of China for financially supporting under Grant No. 11471110, the Foundation Grant of Education Department of Hunan (Nos. 16A126 and 16C0332) and Hunan Provincial Innovation Foundation For Postgraduate under Grant No. CX2017B173.
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Cheng, X., Li, R. & Zhou, Y. Tighter price of anarchy for selfish task allocation on selfish machines. J Comb Optim 44, 1848–1879 (2022). https://doi.org/10.1007/s10878-020-00556-6
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DOI: https://doi.org/10.1007/s10878-020-00556-6