Semi-online scheduling on two uniform machines with the known largest size

Abstract

This paper investigates semi-online scheduling on two uniform machines with the known largest size. Denote by s _j the speed of each machine, j=1,2. Assume 0<s ₁≤s ₂, and let s=s ₂/s ₁ be the speed ratio. First, for the speed ratio $s\in [1,\sqrt{2}]$ , we present an optimal semi-online algorithm $\mathcal{LSMP}$ with the competitive ratio $\mathrm{max}\{\frac {2(s+1)}{2s+1},s\}$ . Second, we present a semi-online algorithm $\mathcal{HSMP}$ . And for $s\in(\sqrt{2},1+\sqrt{3})$ , the competitive ratio of $\mathcal{HSMP}$ is strictly smaller than that of the online algorithm $\mathcal{LS}$ . Finally, for the speed ratio s≥s ^*≈3.715, we show that the known largest size cannot help us to design a semi-online algorithm with the competitive ratio strictly smaller than that of $\mathcal{LS}$ . Moreover, we show a lower bound for $s\in(\sqrt{2},s^{*})$ .