113 research outputs found
Some topics on deterministic scheduling problems
Sequencing and scheduling problems are motivated by allocation of limited resources over time. The goal is to find an optimal allocation where optimality is defined by some problem specific objectives.
This dissertation considers the scheduling of a set of ri tasks, with precedence constraints, on m \u3e= 1 identical and parallel processors so as to minimize the makespan. Specifically, it considers the situation where tasks, along with their precedence constraints, are released at different times, and the scheduler has to make scheduling decisions without knowledge of future releases. Both preemptive and nonpreemptive schedules are considered. This dissertation shows that optimal online algorithms exist for some cases, while for others it is impossible to have one. The results give a sharp boundary delineating the possible and the impossible cases.
Then an O(n log n)-time implementation is given for the algorithm which solves P|pj = 1, rj, outtree| ΣCj and P|pmtn, pj=1,rj,outtree|ΣCj.
A fundamental problem in scheduling theory is that of scheduling a set of n unit-execution-time (UET) tasks, with precedence constraints, on m \u3e 1 parallel and identical processors so as to minimize the mean flow time. For arbitrary precedence constraints, this dissertation gives a 2-approximation algorithm. For intrees, a 1.5-approximation algorithm is given.
Six dual criteria problems are also considered in this dissertation. Two open problems are first solved. Both problems are single machine scheduling problems with the number of tardy jobs as the primary criterion and with the total completion time and the total tardiness as the secondary criterion, respectively. Both problems are shown to be NP-hard. Then it focuses on bi-criteria scheduling problems involving the number of tardy jobs, the maximum weighted tardiness and the maximum tardiness. NP-hardness proofs are given for the scheduling problems when the number of tardy jobs is the primary criterion and the maximum weighted tardiness is the secondary criterion, or vice versa. It then considers complexity relationships between the various problems, gives polynomial-time algorithms for some special cases, and proposes fast heuristics for the general case
Streaming Approximation Scheme for Minimizing Total Completion Time on Parallel Machines Subject to Varying Processing Capacity
We study the problem of minimizing total completion time on parallel machines
subject to varying processing capacity. In this paper, we develop an
approximation scheme for the problem under the data stream model where the
input data is massive and cannot fit into memory and thus can only be scanned
for a few passes. Our algorithm can compute the approximate value of the
optimal total completion time in one pass and output the schedule with the
approximate value in two passes
Multitasking Scheduling with Shared Processing
Recently, the problem of multitasking scheduling has attracted a lot of
attention in the service industries where workers frequently perform multiple
tasks by switching from one task to another. Hall, Leung and Li (Discrete
Applied Mathematics 2016) proposed a shared processing multitasking scheduling
model which allows a team to continue to work on the primary tasks while
processing the routinely scheduled activities as they occur. The processing
sharing is achieved by allocating a fraction of the processing capacity to
routine jobs and the remaining fraction, which we denote as sharing ratio, to
the primary jobs.
In this paper, we generalize this model to parallel machines and allow the
fraction of the processing capacity assigned to routine jobs to vary from one
to another. The objectives are minimizing makespan and minimizing the total
completion time. We show that for both objectives, there is no polynomial time
approximation algorithm unless P=NP if the sharing ratios are arbitrary for all
machines. Then we consider the problems where the sharing ratios on some
machines have a constant lower bound. For each objective, we analyze the
performance of the classical scheduling algorithms and their variations and
then develop a polynomial time approximation scheme when the number of machines
is a constant
Sublinear Approximation Schemes for Scheduling Precedence Graphs of Bounded Depth
We study the classical scheduling problem on parallel machines %with precedence constraints where the precedence graph has the bounded depth h. Our goal is to minimize the maximum completion time. We focus on developing approximation algorithms that use only sublinear space or sublinear time. We develop the first one-pass streaming approximation schemes using sublinear space when all jobs\u27 processing times differ no more than a constant factor c and the number of machines m is at most 2nϵ3hc. This is so far the best approximation we can have in terms of m, since no polynomial time approximation better than 43 exists when m=n3 unless P=NP. %the problem cannot be approximated within a factor of 43 when m=n3 even if all jobs have equal processing time. The algorithms are then extended to the more general problem where the largest αn jobs have no more than c factor difference. % for some constant
Multitasking Scheduling with Shared Processing
Recently, the problem of multitasking scheduling has attracted a lot of attention in the service industries where workers frequently perform multiple tasks by switching from one task to another. Hall, Leung and Li (Discrete Applied Mathematics 2016) proposed a shared processing multitasking scheduling model which allows a team to continue to work on the primary tasks while processing the routinely scheduled activities as they occur. The processing sharing is achieved by allocating a fraction of the processing capacity to routine jobs and the remaining fraction, which we denote as sharing ratio, to the primary jobs. In this paper, we generalize this model to parallel machines and allow the fraction of the processing capacity assigned to routine jobs to vary from one to another. The objectives are minimizing makespan and minimizing the total completion time. We show that for both objectives, there is no polynomial time approximation algorithm unless P = NP if the sharing ratios are arbitrary for all machines. Then we consider the problems where the sharing ratios on some machines have a constant lower bound. For each objective, we analyze the performance of the classical scheduling algorithms and their variations and then develop a polynomial time approximation scheme when the number of machines is a constant
Streaming Approximation Scheme for Minimizing Total Completion Time on Parallel Machines Subject to Varying Processing Capacity
We study the problem of minimizing total completion time on parallel machines subject to varying processing capacity. In this paper, we develop an approximation scheme for the problem under the data stream model where the input data is massive and cannot fit into memory and thus can only be scanned for a few passes. Our algorithm can compute the approximate value of the optimal total completion time in one pass and output the schedule with the approximate value in two passes
Streaming algorithms for multitasking scheduling with shared processing
In this paper, we design the first streaming algorithms for the problem of multitasking scheduling on parallel machines with shared processing. In one pass, our streaming approximation schemes can provide an approximate value of the optimal makespan. If the jobs can be read in two passes, the algorithm can find the schedule with the approximate value. This work not only provides an algorithmic big data solution for the studied problem, but also gives an insight into the design of streaming algorithms for other problems in the area of scheduling
Streaming Algorithms for Multitasking Scheduling with Shared Processing
In this paper, we design the first streaming algorithms for the problem of
multitasking scheduling on parallel machines with shared processing. In one
pass, our streaming approximation schemes can provide an approximate value of
the optimal makespan. If the jobs can be read in two passes, the algorithm can
find the schedule with the approximate value. This work not only provides an
algorithmic big data solution for the studied problem, but also gives an
insight into the design of streaming algorithms for other problems in the area
of scheduling
A fast preemptive scheduling algorithm with release times and inclusive processing set restrictions
AbstractWe consider the problem of preemptively scheduling n independent jobs on m parallel machines so as to minimize the makespan. Each job Jj has a release time rj and it can only be processed on a subset of machines Mj. The machines are linearly ordered. Each job Jj has a machine index aj such that Mj={Maj,Maj+1,…,Mm}. We first show that there is no 1-competitive online algorithm for this problem. We then give an offline algorithm with a running time of O(nklogP+mnk2+m3k), where k is the number of distinct release times and P is the total processing time of all jobs
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