Faster algorithms for bicriteria scheduling of identical jobs on uniform machines

Abstract

<p style='text-indent:20px;'>This paper studies the problem of scheduling <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> jobs with equal processing times on <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> uniform machines to optimize two criteria simultaneously. The main contribution is an <inline-formula><tex-math id="M3">\begin{document}$ O(n\log m+n^3) $\end{document}</tex-math></inline-formula>-time algorithm for two general min-max criteria, improving the previous <inline-formula><tex-math id="M4">\begin{document}$ O(n\log m+n^4) $\end{document}</tex-math></inline-formula> time complexity. For a particular min-sum criterion (total weighted completion time or total tardiness) in combination with a general min-max criterion, <inline-formula><tex-math id="M5">\begin{document}$ O(n\log m+n^3) $\end{document}</tex-math></inline-formula>-time algorithms are also obtained, improving the previous <inline-formula><tex-math id="M6">\begin{document}$ O(n\log m+n^3\log n) $\end{document}</tex-math></inline-formula> time complexity. The algorithms can produce all Pareto optimal points together with the corresponding schedules.</p>