SRPT is 1.86-Competitive for Completion Time Scheduling

Abstract

We consider the classical problem of scheduling preemptible jobs, that arrive over time, on identical parallel machines. The goal is to minimize the total completion time of the jobs. In standard scheduling notation of Graham et al. [5], this problem is denoted P | rj, pmtn | P j cj . A popular algorithm called SRPT, which always schedules the unfinished jobs with shortest remaining processing time, is known to be 2-competitive, see Phillips et al. [13,14]. This is also the best known competitive ratio for any online algorithm. However, it is conjectured that the competitive ratio of SRPT is significantly less than 2. Even breaking the barrier of 2 is considered a significant step towards the final answer of this classical online problem. We improve on this open problem by showing that SRPT is 1.86-competitive. This result is obtained using the following method, which might be of general interest: We define two dependent random variables that sum up to the difference between the cost of an SRPT schedule and the cost of an optimal schedule. Then we bound the sum of the expected values of these random variables with respect to the cost of the optimal schedule, yielding the claimed competitiveness. Furthermore, we show a lower bound of 21/19 for SRPT, improving on the previously best known 12/11 due to Lu et al. [11].

Completion Time Scheduling and the WSRPT Algorithm

“…We also provide improved general lower bounds of 21/19 ≈ 1.105 and 1.114 on the competitive ratio of any deterministic algorithm for the problems P | r j , pmtn | j c j and P | r j , pmtn | j w j c j , respectively. We believe that 21/19 is the correct answer for the unweighted problem, as we agree with the conjecture of [11] that the algorithm SRPT is 21/19-competitive.…”

“…We prove these lower bounds by giving a simple scheme for constructing an instance where the nature of the jobs that arrive depends on the choices the algorithm has made so far. The scheme is based on the lower bound instance for the algorithm SRPT, from [11]. Theorem 1.…”

“…It was a long-held belief that SRPT was close to optimal, and most recently, [6] finally showed that SRPT is 5/4-competitive. (Prior to that, the original proof of SRPT's 2-competitiveness [7] was the best known ratio for fifteen years until [11] showed it was 1.86-competitive.) The current lowerbound on the competitiveness of SRPT is 21/19 ≈ 1.105 [11].…”

Efficient Algorithms for Average Completion Time Scheduling

“…Hence, a new idea is needed to prove a smaller ratio. Indeed, the proof by Chung et al [5] is completely different and uses a sophisticated randomized analysis of the optimal solution. On the contrary, our proof builds on the original proof of Phillips et al and continues where that proof stops.…”

“…The example in Figure 1 shows that this is not true when SRPT is applied to parallel machines. The best known upper bound on its competitive ratio was 2 [16] until recently (SODA2010), Chung et al [5] showed that the ratio is at most 1.86. Moreover, they show that the ratio is not better than 21/19 > 1.105.…”