Approximation algorithms for the parallel flow shop problem

We consider the NP-hard problem of scheduling n jobs in m two-stage parallel flow shops so as to minimize the makespan. This problem decomposes into two subproblems: assigning the jobs to parallel flow shops; and scheduling the jobs assigned to the same flow shop by use of Johnson's rule. For m = 2, we present a 32-approximation algorithm, and for m = 3, we present a 127-approximation algorithm. Both these algorithms run in O(n log n) time. These are the first approximation algorithms with fixed worst-case performance guarantees for the parallel flow shop problem

Scheduling MapReduce Jobs under Multi-Round Precedences

We consider non-preemptive scheduling of MapReduce jobs with multiple tasks in the practical scenario where each job requires several map-reduce rounds. We seek to minimize the average weighted completion time and consider scheduling on identical and unrelated parallel processors. For identical processors, we present LP-based O(1)-approximation algorithms. For unrelated processors, the approximation ratio naturally depends on the maximum number of rounds of any job. Since the number of rounds per job in typical MapReduce algorithms is a small constant, our scheduling algorithms achieve a small approximation ratio in practice. For the single-round case, we substantially improve on previously best known approximation guarantees for both identical and unrelated processors. Moreover, we conduct an experimental analysis and compare the performance of our algorithms against a fast heuristic and a lower bound on the optimal solution, thus demonstrating their promising practical performance

Developing graph-based co-scheduling algorithms on multicore computers

It is common that multiple cores reside on the same chip and share the on-chip cache. As a result, resource sharing can cause performance degradation of co-running jobs.Job co-scheduling is a technique that can effectively alleviate this contention and many co-schedulers have been reported in related literature. Most solutions however do not aim to find the optimal co-scheduling solution. Being able to determine the optimal solution is critical for evaluating co-scheduling systems. Moreover, most co-schedulers only consider serial jobs, and there often exist both parallel and serial jobs in real-world systems. In this paper a graph-based method is developed to find the optimal co-scheduling solution for serial jobs; the method is then extended to incorporate parallel jobs, including multi-process, and multithreaded parallel jobs. A number of optimization measures are also developed to accelerate the solving process. Moreover, a flexible approximation technique is proposed to strike a balance between the solving speed and the solution quality. Extensive experiments are conducted to evaluate the effectiveness of the proposed co-scheduling algorithms. The results show that the proposed algorithms can find the optimal co-scheduling solution for both serial and parallel jobs. The proposed approximation technique is also shown to be flexible in the sense that we can control the solving speed by setting the requirement for the solution quality

Tight Approximation Algorithms for Scheduling with Fixed Jobs and Non-Availability

We study two closely related problems in non-preemptive scheduling of sequential jobs on identical parallel machines. In these two settings there are either fixed jobs or non-availability intervals during which the machines are not available; in both cases, the objective is to minimize the makespan. Both formulations have different applications, e.g. in turnaround scheduling or overlay computing. For both problems we contribute approximation algorithms with an improved ratio of $3/2+\epsilon$, respectively, which we refine to approximation algorithms with ratio $3/2$. For scheduling with fixed jobs, a lower bound of $3/2$ on the approximation ratio has been obtained by Scharbrodt, Steger & Weisser: for scheduling with non-availability we provide the same lower bound. In total, our approximation ratio for both problems is tight via suitable inapproximability results. We use dual approximation, creation of a gap structure and job configurations, and a PTAS for the multiple subset sum problem. However, the main feature of our algorithms is a new technique for the assignment of large jobs via flexible rounding. Our new technique is based on an interesting cyclic shifting argument in combination with a network flow model for the assignment of jobs to large gaps

Tight Approximation Algorithms for Scheduling with Fixed Jobs and Non-Availability

We study two closely related problems in non-preemptive scheduling of jobs on identical parallel machines. In these two settings there are either fixed jobs or non-availability intervals during which the machines are not available; in both cases, the objective is to minimize the makespan. Both formulations have different applications, e.g. in turnaround scheduling or overlay computing. For both problems we contribute approximation algorithms with an improved ratio of 3/2. For scheduling with fixed jobs, a lower bound of 3/2 on the approximation ratio has been obtained by Scharbrodt, Steger & Weisser; for scheduling with non-availability we provide the same lower bound. We use dual approximation, creation of a gap structure and a PTAS for the multiple subset sum problem, combined with a post- processing step to assign large job

Approximation Algorithms for Scheduling Parallel Jobs: Breaking the Approximation Ratio of 2

In this paper we study variants of the non-preemptive parallel job scheduling problem in which the number of machines is polynomially bounded in the number of jobs. For this problem we show that a schedule with length at most (1 + ε)OPT can be calculated in polynomial time. Unless P = NP, this is the best possible result (in the sense of approximation ratio), since the problem is strongly NP-hard. For the case, where all jobs must be allotted to a subset of consecutive machines, a schedule with length at most (1.5 + ε)OPT can be calculated in polynomial time. The previously best known results are algorithms with absolute approximation ratio 2. Furthermore, we extend both algorithms to the case of malleable jobs with the same approximation ratios

Machine Scheduling with Resource Dependent Processing Times

We consider several parallel machine scheduling settings with the objective to minimize the schedule makespan. The most general of these settings is unrelated parallel machine scheduling. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource. A given amount of that resource, e.g. workers, can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes classical machine scheduling problems, adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of integer programming formulations for relaxations of the respective problems, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham''s list scheduling, we thus prove the existence of constant factor approximation algorithms. Our performance guarantee is 6.83 for the most general case of unrelated parallel machine scheduling. We improve this bound for two special cases, namely to 5.83 whenever the jobs are assigned to machines beforehand, and to (5+e), e>0, whenever the processing times do not depend on the machine. Moreover, we discuss tightness of the relaxations, and derive inapproximability results.operations research and management science;

Approximation Techniques for Average Completion Time Scheduling

We consider the problem of nonpreemptive scheduling to minimize average ( weighted) completion time, allowing for release dates, parallel machines, and precedence constraints. Recent work has led to constant-factor approximations for this problem based on solving a preemptive or linear programming relaxation and then using the solution to get an ordering on the jobs. We introduce several new techniques which generalize this basic paradigm. We use these ideas to obtain improved approximation algorithms for one-machine scheduling to minimize average completion time with release dates. In the process, we obtain an optimal randomized on-line algorithm for the same problem that beats a lower bound for deterministic on-line algorithms. We consider extensions to the case of parallel machine scheduling, and for this we introduce two new ideas: first, we show that a preemptive one-machine relaxation is a powerful tool for designing parallel machine scheduling algorithms that simultaneously produce good approximations and have small running times; second, we show that a nongreedy “rounding” of the relaxation yields better approximations than a greedy one. We also prove a general theore mrelating the value of one- machine relaxations to that of the schedules obtained for the original m-machine problems. This theorem applies even when there are precedence constraints on the jobs. We apply this result to obtain improved approximation ratios for precedence graphs such as in-trees, out-trees, and series-parallel graphs