Approximating the Optimal Algorithm for Online Scheduling Problems via Dynamic Programming

Lin Chen (), Deshi Ye () and Guochuan Zhang ()
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Lin Chen: College of Computer Science, Zhejiang University, Hangzhou, 310027, P. R. China
Deshi Ye: College of Computer Science, Zhejiang University, Hangzhou, 310027, P. R. China
Guochuan Zhang: College of Computer Science, Zhejiang University, Hangzhou, 310027, P. R. China

Asia-Pacific Journal of Operational Research (APJOR), 2015, vol. 32, issue 01, 1-19

Abstract: Very recently Günther et al. [E. Günther, O. Maurer, N. Megow and A. Wiese (2013). A new approach to online scheduling: Approximating the optimal competitive ratio. In Proc. 24th Annual ACM-SIAM Symp. Discrete Algorithms (SODA).] initiate a new systematic way of studying online problems by introducing the competitive ratio approximation scheme (simplified as competitive schemes in this paper), which is a class of algorithms {Aϵ|ϵ > 0} with a competitive ratio at most ρ*(1 + ϵ), where ρ* is the best possible competitive ratio over all online algorithms. Along this line, Günther et al. [E. Günther, O. Maurer, N. Megow and A. Wiese (2013). A new approach to online scheduling: Approximating the optimal competitive ratio. In Proc. 24th Annual ACM-SIAM Symp. Discrete Algorithms (SODA).] provide competitive schemes for several online over time scheduling problems like Qm|rj, (pmtn)|∑wjcj, while the running times are polynomial if the number of machines is a constant. In this paper, we consider the classical online scheduling problems, where jobs arrive in a list. We present competitive schemes for Rm‖Cmax and $Rm\Vert \sum_i C_i^p$, where the running times are polynomial if the number of machines is a constant. Specifically, we are able to derive a competitive scheme for P‖Cmax which runs in polynomial time even if the number of machines is an input. Our method is novel and more efficient than that of Günther et al. [E. Günther, O. Maurer, N. Megow and A. Wiese (2013). A new approach to online scheduling: Approximating the optimal competitive ratio. In Proc. 24th Annual ACM-SIAM Symp. Discrete Algorithms (SODA).] Indeed, by utilizing the standard rounding technique for the off-line scheduling problems, we reformulate the online scheduling problem into a game on an infinite graph, through which we arrive at the following key point: Assuming that the best competitive ratio is ρ*, for any online algorithm there exists a list of a polynomial number of jobs showing that its competitive ratio is at least ρ* - O(ϵ). Interestingly such a result is achieved via a dynamic programming algorithm. Our framework is also applicable to other online problems.

Keywords: Competitive analysis; online scheduling; dynamic programming (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1142/S0217595915400114

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