Semi-online scheduling jobs with tightly-grouped processing times on three identical machines

This paper investigates the semi-online version of scheduling problem P||Cmax on a three-machine system. We assume that all jobs have their processing times between p and rp (p>0,r⩾1). We give a comprehensive competitive ratio of LS algorithm which is a piecewise function on r⩾1. It shows that LS is an optimal semi-online algorithm for every r∈[1,1.5], [3,2] and [6,+∞). We further present an optimal algorithm for every r∈[2,2.5], and an almost optimal algorithm for every r∈(2.5,3] where the largest gap between its competitive ratio and the lower bound of the problem is at most 0.01417. We also present an improved algorithm with smaller competitive ratio than that of LS for every r∈(3,6).