LPT online strategy for parallel-machine scheduling with kind release times

Abstract

In this paper, we introduce a new environment of online scheduling in which jobs have kind release times (KRT). Under the KRT assumption, no jobs can be released when all of the machines are busy. Then we study the competitive ratio of the online LPT algorithm to minimize makespan on $m$ identical parallel machines under the KRT assumption. The online LPT algorithm can be described as follows: At any time a machine becomes available for processing, schedule an available job with the largest processing time. When $m=2$, we show that LPT is a best possible online algorithm with a competitive ratio of $5/4$. For each $m\ge 3$, we show that LPT has a competitive ratio of $3/2$. We also show that LPT is a best possible online dense-algorithm, where an online algorithm is called dense if it always immediately assigns currently available jobs to one of the idle machines as long as there are at least two idle machines. Furthermore, for $m\ge 3$, we show that no online algorithms have a competitive ratio less than $1.3473$. Comparing to the known result of Chen and Vestjens (Operations Research Letters 1997), which establishes the $3/2$-competitiveness of LPT, the results in this paper imply that the KRT assumption can help to improve the competitive ratio of LPT for two parallel machines.