Abstract: The first fully polynomial-time approximation algorithm for solving the Job Shop problem in its most general form with theoretically guaranteed a priori accuracy bounds was published in (Sevast'yanov, 1984). The approach used in that paper for solving the Job Shop problem, based on the method of compact vector summation in $d$-dimensional space, was further developed in (Sevast'yanov, 1986). For any fixed job shop, those algorithms provided an asymptotic optimality of the solutions, under the unlimited growth of size of the batch of jobs supplied at the problem input. Later on, two other approximation algorithms for this problem with ratio performance guarantees appeared in (Shmoys et al., 1994). The first algorithm provided (in polynomial time) a poly-logarithmic approximation in terms of two parameters: the number of machines ($m$) and the number of operations per job ($\mu$). The second algorithm guaranteed a $(2+\varepsilon)$-approximation for any fixed $\varepsilon>0$ and was polynomial-time for any fixed values of $m$ and $\mu$. Finally, in (Jansen et al., 2003), a PTAS was elaborated for the Job Shop problem, which was polynomial-time under the assumption that $m$ and $\mu$ are fixed. All three above mentioned algorithms were heavily based on the algorithm from (Sevast'yanov, 1986) and were presented as an ``improvement'' of the latter. In the present paper, a comparative analysis of the quality of four known for today theoretical algorithms for approximate solution of the Job Shop problem (namely, of the algorithms from (Sevast'yanov, 1986), (Shmoys et al., 1994), and (Jansen et al., 2003)) is carried out. It is shown that on the set of instances of practical size, the best to date by the accuracy/efficiency criteria is the algorithm from (Sevast'yanov, 1986), while the second algorithm from (Shmoys et al., 1994) and the PTAS from (Jansen et al., 2003) are not capable of obtaining a solution 10 times worse than optimal in any physically observable time even for comparatively small instances of the problem (and even with the help of the Super-Computer System of the whole Universe). The paper also presents an improved scheme of the algorithm from (Sevast'yanov, 1986), which gives a better bound on the absolute error with the same bound on the running time, which is the first positive result in this direction since 1986. Detailed schemes of all procedures used in the algorithm are given (in order to facilitate its programming). In addition, the paper shows how both algorithms from (Shmoys et al., 1994) can be transformed so that their running time becomes linear in the main problem parameter, the number of jobs $n$.

Cite: Sevastyanov S.
Approximate solution of the job shop problem by means of the compact vector summation within a bounded convex set in R^d
Сибирские электронные математические известия (Siberian Electronic Mathematical Reports). 2025.

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