Lower bounds on online deadline scheduling with preemption penalties

We generalize and improve previous results of online preemptive deadline scheduling with preemption penalties. We consider both the preemption-restart and the preemption-resume models, and give new or improved lower bounds on the competitive ratio of deterministic online algorithms. In many cases the bounds are optimal when the job deadlines are tight. Our results show that the competitiveness varies linearly with the penalty factor.     

Online interval scheduling on a single machine with finite lookahead

We study an online weighted interval scheduling problem on a single machine, where all intervals have unit length and the objective is to maximize the total weight of all completed intervals. We investigate how the function of finite lookahead improves the competitivities of deterministic online heuristics, under both preemptive and non-preemptive models. The lookahead model studied in this paper is that an online heuristic is said to have a lookahead ability of LD if at any time point it is able to foresee all the intervals to be released within the next LD   units of time. We investigate both competitive online heuristics and lower bounds on the competitive ratio, with lookahead 0≤LD≤1$0\le \mathit{LD}\le 1$ under the preemptive model, and lookahead 0≤LD≤2$0\le \mathit{LD}\le 2$ under the non-preemptive model. A method to transform a preemptive lookahead online algorithm to a non-preemptive online algorithm with enhanced lookahead ability is also given. Computational tests are performed to compare the practical competitivities of the online heuristics with different lookahead abilities.     

Online deadline scheduling on faster machines

Online deadline scheduling is to determine which jobs are accepted or rejected, where jobs have the deadline by which they must finish their processing and they arrive in the online fashion. The slack of a job is the gap between its arrival time and the last time when it can first be scheduled to meet its deadline. The job instance is given such that the slack of each job is at least κ times its processing time, where κ is called patience. In this paper, online algorithms have faster machines than the adversary. We investigate the speed of machines on which the online algorithms can achieve the optimality and parametrize them by the patience.     

Improved on-line broadcast scheduling with deadlines

We study an on-line broadcast scheduling problem in which requests have deadlines, and the objective is to maximize the weighted throughput, i.e., the weighted total length of the satisfied requests. For the case where all requested pages have the same length, we present an online deterministic algorithm named BAR and prove that it is 4.56-competitive. This improves the previous algorithm of (Kim, J.-H., Chwa, K.-Y. in Theor. Comput. Sci. 325(3):479–488, 2004) which is shown to be 5-competitive by (Chan, W.-T., et al. in Lecture Notes in Computer Science, vol. 3106, pp. 210–218, 2004). In the case that pages may have different lengths, we give a ( )-competitive algorithm where Δ is the ratio of maximum to minimum page lengths. This improves the (4Δ+3)-competitive algorithm of (Chan, W.-T., et al. in Lecture Notes in Computer Science, vol. 3106, pp. 210–218, 2004). We also prove an almost matching lower bound of Ω(Δ/log Δ). Furthermore, for small values of Δ we give better lower bounds. The work described in this paper was fully supported by grants from the Research Grants Council of the Hong Kong SAR, China [CityU 1198/03E, HKU 7142/03E, HKU 5172/03E], an NSF Grant of China [No. 10371094], and a Nuffield Foundation Grant of UK [NAL/01004/G].     

Optimal online-list batch scheduling

We consider the online-list batch scheduling problem. Jobs arrive one by one and have to be assigned upon arrival to a scheduled batch such that the makespan is minimized. Each batch can accommodate up to B jobs. We give a complete classification of the tractability of this online problem.     

Online scheduling with preemption or non-completion penalties

We consider online preemptive scheduling problems where jobs have deadlines and the objective is to maximize the total weight of jobs completed before their deadlines. In the first problem, preemptions are not free but incur a penalty. In the second problem, a job has to be accepted or rejected immediately upon arrival, and may need to be immediately allocated a fixed scheduling interval as well; if these accepted jobs are not eventually completed, the job is lost, and a penalty is incurred. We give an algorithm with the optimal competitive ratio for the first problem, and new and improved algorithms for the second problem, under different models of preemptions and job weights.     

Scheduling with job release dates, delivery times and preemption penalties

We consider in this paper the single-machine preemptive scheduling problem with job release dates, delivery times and preemption penalties, where each time a job is started, whether initially or after preemption, a job-dependent setup must take place. First, we prove that the problem is strongly NP-hard. Then, we present a dynamic programming algorithm and a polynomial time approximation scheme.     

Online scheduling on semi-related machines

We study the problem of minimizing the makespan on related machines in the following setting: jobs arrive over time, and the machines may become available or unavailable. In either case, advance warning is given, i.e., the next point of time where a job is released or a machine changes state is revealed a little time ahead. We present an optimal online algorithm for this problem.     

Energy-efficient deadline scheduling for heterogeneous systems

Energy efficiency is a major concern in modern high performance computing (HPC) systems and a power-aware scheduling approach is a promising way to achieve that. While there are a number of studies in power-aware scheduling by means of dynamic power management (DPM) and/or dynamic voltage and frequency scaling (DVFS) techniques, most of them only consider scheduling at a steady state. However, HPC applications like scientific visualization often need deadline constraints to guarantee timely completion. In this paper we present power-aware scheduling algorithms with deadline constraints for heterogeneous systems. We formulate the problem by extending the traditional multiprocessor scheduling and design approximation algorithms with analysis on the worst-case performance. We also present a pricing scheme for tasks in the way that the price of a task varies as its energy usage as well as largely depending on the tightness of its deadline. Last we extend the proposed algorithm to the control dependence graph and the online case which is more realistic. Through the extensive experiments, we demonstrate that the proposed algorithm achieves near-optimal energy efficiency, on average 16.4% better for synthetic workload and 12.9% better for realistic workload than the EDD (Earliest Due Date)-based algorithm; The extended online algorithm also outperforms the EDF (Earliest Deadline First)-based algorithm with an average up to 26% of energy saving and 22% of deadline satisfaction. It is experimentally shown as well that the pricing scheme provides a flexible trade-off between deadline tightness and price.     

Two processor scheduling with limited preemption

We introduce deterministic task system scheduling with limited preemption—that is, a preempted task can resume execution only on the processor where it originally executed. Two classes of limited preemptive schedules are introduced, corresponding to whether or not unforced idle time is allowed in the schedule. Our main result is to show that, on two processors, these two classes are equivalent with respect to optimal schedule lengths. We use this result to establish a hierarchy of optimal schedule lengths.     

Online scheduling of two job types on a set of multipurpose machines with unit processing times

We study a problem of scheduling a set of n jobs with unit processing times on a set of m multipurpose machines in which the objective is to minimize the makespan. It is assumed that there are two different job types, where each job type can be processed on a unique subset of machines. We provide an optimal offline algorithm to solve the problem in constant time and an online algorithm with a competitive ratio that equals the lower bound. We show that the worst competitive ratio is obtained for an inclusive job-machine structure in which the first job type can be processed on any of the m machines while the second job type can be processed only on a subset of m/2 machines. Moreover, we show that our online algorithm is 1-competitive if the machines are not flexible, i.e., each machine can process only a single job type.     

Online traveling salesman problem with deadline and advanced information

We consider the online version of the traveling salesman problem, where instances are not known in advance. Requests are released over time regardless whether the server is en route or not. This problem has been described as online TSP. Current literature about online TSP assumes that each request becomes known at its release time and will always remain active. We model the customers’ waiting psychology and service preparation time into the online TSP with the objective to serve as many requests as possible. More specifically, each request has a disclosure time before accepting service at its release time, and a deadline, which is no bigger than its release time plus the travel time from origin to its position. We give lower bounds for the competitive ratios, online algorithms, and quantify the influence of advanced information on competitive ratios.     

Online scheduling with rejection and withdrawal

We study an online scheduling problem with rejection, in which some rearrangement of the solution is allowed. This problem is called scheduling with rejection and withdrawal. Each arriving job has a processing time and a rejection cost associated with it, and it needs to be either assigned to a machine or rejected upon arrival. At termination, it is possible to choose at most a fixed number of scheduled jobs and withdraw them (i.e., decide to reject them). We study the minimization version, where the goal is to minimize the sum of the makespan and the total rejection cost (which corresponds to the penalty), and the maximization problem, where the goal is to maximize the sum of the minimum load and the total rejection cost (which corresponds to profit). We study environments of machines, which are the case of m identical machines and the case of two uniformly related machines, and show a strong relation between these problems and the related classic online scheduling problems which they generalize, in contrast to standard scheduling with rejection, which typically makes the scheduling problems harder.     

A new algorithm for online uniform-machine scheduling to minimize the makespan

We consider the online scheduling problem with m−1, m?2, uniform machines each with a processing speed of 1, and one machine with a speed of s, 1?s?2, to minimize the makespan. The well-known list scheduling (LS) algorithm has a worst-case bound of [Y. Cho, S. Sahni, Bounds for list schedules on uniform processors, SIAM J. Comput. 9 (1980) 91-103]. An algorithm with a better competitive ratio was proposed in [R. Li, L. Shi, An on-line algorithm for some uniform processor scheduling, SIAM J. Comput. 27 (1998) 414-422]. It has a worst-case bound of 2.8795 for a big m and s=2. In this note we present a 2.45-competitive algorithm for m?4 and any s, 1?s?2.     

Online Randomized Multiprocessor Scheduling

The use of randomization in online multiprocessor scheduling is studied. The problem of scheduling independent jobs on m machines online originates with Graham [16]. While the deterministic case of this problem has been studied extensively, little work has been done on the randomized case. For m= 2 a randomized 4/3-competitive algorithm was found by Bartal et al. A randomized algorithm for m ≥ 3 is presented. It achieves competitive ratios of 1.55665, 1.65888, 1.73376, 1.78295, and 1.81681, for m = 3, 4, 5, 6,7 , respectively. These competitive ratios are less than the best deterministic lower bound for m=3,4,5 and less than the best known deterministic competitive ratio for m = 3,4,5,6,7 . Two models of multiprocessor scheduling with rejection are studied. The first is the model of Bartal et al. Two randomized algorithms for this model are presented. The first algorithm performs well for small m , achieving competitive ratios of 3/2 , , for m=2,3,4 , respectively. The second algorithm outperforms the first for m ≥ 5 . It beats the deterministic algorithm of Bartal et al. for all m = 5 ,. . ., 1000 . It is conjectured that this is true for all m . The second model differs in that preemption is allowed. For this model, randomized algorithms are presented which outperform the best deterministic algorithm for small m . Received August 11, 1997; revised February 25, 1998.     

Online scheduling on unbounded parallel-batch machines to minimize maximum flow-time

We consider online scheduling on m unbounded parallel-batch machines to minimize maximum flow-time of the jobs. We show that no online algorithm can have a competitive ratio less than 1+αm, where αm is the positive root of α²+(m+1)α−1=0, and this lower bound is still valid even when all jobs have the same processing times. Then we provide an online algorithm of competitive ratio 1+1/m. When the jobs have the same processing times, we present a best possible online algorithm of competitive ratio 1+αm.