A tight linear time 1312\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{13}{12}$$\end{document}-approximation algorithm for the P2||Cmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P2 || C_{\max }$$\end{document} problem

We consider problem P2||Cmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P2 || C_{\max }$$\end{document} where the goal is to schedule n jobs on two identical parallel machines to minimize the makespan. We focus on constant factor approximation algorithms with complexity independent from the required accuracy. We exploit the famous Longest Processing Time (LPT) rule that requires to sort jobs in non-ascending order of processing times and then to assign one job at a time to the machine whose load is smallest so far. We propose an approximation algorithm that applies LPT to a subset of 2k jobs, then considers a single step of local search on the obtained subschedule and finally applies list scheduling to the remaining jobs. This algorithm, when applied for k=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=5$$\end{document}, reaches a tight 1312\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{13}{12}$$\end{document}-approximation ratio improving the ratio of 1211\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{12}{11}$$\end{document} proposed by He et al. (Nav Res Logist 47:593–601, 2000). We use Mathematical Programming to analyze the approximation ratio of our approach. As a byproduct, we also show how to improve the FPTAS of Sahni for any n>1/ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n > 1/\epsilon $$\end{document} so as to reach an approximation ratio (1+ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1 + \epsilon )$$\end{document} with time complexity O(n+1ϵ3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n + \frac{1}{\epsilon ^3})$$\end{document}.