Estimating the Makespan of the Two-Valued Restricted Assignment Problem

We consider a special case of the scheduling problem on unrelated machines, namely the restricted assignment problem with two different processing times. We show that the configuration LP has an integrality gap of at most 53≈1.667\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{5}{3} \approx 1.667$$\end{document} for this problem. This allows us to estimate the optimal makespan within a factor of 53\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{5}{3}$$\end{document}, improving upon the previously best known estimation algorithm with ratio 116≈1.833\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{11}{6} \approx 1.833$$\end{document} due to Chakrabarty et al. (in: Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms (SODA 2015), pp 1087–1101, 2015).