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 with Time Lags

Scheduling is essential when activities need to be allocated to scarce resources over time. Motivated by the problem of scheduling barges along container terminals in the Port of Rotterdam, this thesis designs and analyzes algorithms for various on-line and off-line scheduling problems with time lags. A time lag specifies a minimum time delay required between the execution of two consecutive operations of the same job. Time lags may be the result of transportation delays (like the time required for barges to sail from one terminal to the next), the duration of activities that do not require resources (like drying or cooling down), or intermediate processes on non-bottleneck machines between two bottleneck machines. For the on-line flow shop, job shop and open shop problems of minimizing the makespan, we analyze the competitive ratio of a class of greedy algorithms. For the off-line parallel flow shop scheduling problem with time lags of minimizing the makespan, we design algorithms with fixed worst-case performance guarantees. For two special subsets of scheduling problems with time lags, we show that Polynomial-Time Approximation Schemes (PTAS) can be constructed under certain mild conditions. For the fixed interval scheduling problem, we show that the flow shop problem is solvable in polynomial time in the case of equal time lags but that it is NP-hard in the strong sense for general time lags. The fixed interval two-machine job shop and open shop problems are shown to be solvable in polynomial time if the time lags are smaller than the processing time of any operation

The Coupled Task Scheduling Problem: Models and Solution Methods

University of Technology Sydney. Faculty of Science.The coupled task scheduling problem (CTSP) is studied in this thesis. The problem consists of scheduling a set of jobs on one or a set of machines, where each job consists of at least two tasks. The main characteristic of the problem is a fixed time-lag between the process of each two consecutive tasks of the same job, where its duration is fixed, i.e., the succeeding task cannot be started earlier or later than the time-lag is passed. The fixed time-lags were introduced to model radar tracking systems, and later extended to formulate problems in chemistry manufacturing systems and robotic cells. The motivation for studying the CTSP in this thesis is to model certain problems in healthcare scheduling with the same characteristics. One example is the scheduling of patients in a chemotherapy clinic, where each patient must undergo a number of consecutive treatments with time-lags in between. Meeting the fixed delays between the treatments of a patient is an important factor in gaining the best outcomes for them. To study the CTSP, a literature review is first conducted, followed by studying the problem in different scheduling environments, including the single-machine, parallel-machine, open-shop and flow-shop settings, where we propose several new complexity results and solution algorithms for different variants of the problem. Regarding the single-machine coupled task problem, a new mathematical formulation and two matheuristic algorithms are proposed for the classical problem, as well as a dynamic programming algorithm for a variant of the problem with time-dependent processing times. With regard to the parallel-machine environment, we first explore the complexity of the problem and propose NP-hardness proofs for certain cases, followed by approximation bounds for the two-machine problem. The latter result is then extended to the open-shop scheduling environment. The problem in the flow-shop environment is then extensively investigated under the permutation setting, and also under the case of ordered processing times. A set of publicly available hard data set and state-of-the-art algorithms are proposed for the ordered flow-shops. Then, flow-shop problem with coupled tasks is studied and polynomial-time algorithms are proposed for various settings of the problem, including the ordered processing times

Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc

Asymptotically Optimal Approximation Algorithms for Coflow Scheduling

Many modern datacenter applications involve large-scale computations composed of multiple data flows that need to be completed over a shared set of distributed resources. Such a computation completes when all of its flows complete. A useful abstraction for modeling such scenarios is a {\em coflow}, which is a collection of flows (e.g., tasks, packets, data transmissions) that all share the same performance goal. In this paper, we present the first approximation algorithms for scheduling coflows over general network topologies with the objective of minimizing total weighted completion time. We consider two different models for coflows based on the nature of individual flows: circuits, and packets. We design constant-factor polynomial-time approximation algorithms for scheduling packet-based coflows with or without given flow paths, and circuit-based coflows with given flow paths. Furthermore, we give an $O(\log n/\log \log n)$-approximation polynomial time algorithm for scheduling circuit-based coflows where flow paths are not given (here $n$ is the number of network edges). We obtain our results by developing a general framework for coflow schedules, based on interval-indexed linear programs, which may extend to other coflow models and objective functions and may also yield improved approximation bounds for specific network scenarios. We also present an experimental evaluation of our approach for circuit-based coflows that show a performance improvement of at least 22% on average over competing heuristics.Comment: Fixed minor typo

Experimental Analysis of Algorithms for Coflow Scheduling

Modern data centers face new scheduling challenges in optimizing job-level performance objectives, where a significant challenge is the scheduling of highly parallel data flows with a common performance goal (e.g., the shuffle operations in MapReduce applications). Chowdhury and Stoica introduced the coflow abstraction to capture these parallel communication patterns, and Chowdhury et al. proposed effective heuristics to schedule coflows efficiently. In our previous paper, we considered the strongly NP-hard problem of minimizing the total weighted completion time of coflows with release dates, and developed the first polynomial-time scheduling algorithms with O(1)-approximation ratios. In this paper, we carry out a comprehensive experimental analysis on a Facebook trace and extensive simulated instances to evaluate the practical performance of several algorithms for coflow scheduling, including the approximation algorithms developed in our previous paper. Our experiments suggest that simple algorithms provide effective approximations of the optimal, and that the performance of our approximation algorithms is relatively robust, near optimal, and always among the best compared with the other algorithms, in both the offline and online settings.Comment: 29 pages, 8 figures, 11 table

Efficient heuristics for the parallel blocking flow shop scheduling problem

We consider the NP-hard problem of scheduling n jobs in F identical parallel flow shops, each consisting of a series of m machines, and doing so with a blocking constraint. The applied criterion is to minimize the makespan, i.e., the maximum completion time of all the jobs in F flow shops (lines). The Parallel Flow Shop Scheduling Problem (PFSP) is conceptually similar to another problem known in the literature as the Distributed Permutation Flow Shop Scheduling Problem (DPFSP), which allows modeling the scheduling process in companies with more than one factory, each factory with a flow shop configuration. Therefore, the proposed methods can solve the scheduling problem under the blocking constraint in both situations, which, to the best of our knowledge, has not been studied previously. In this paper, we propose a mathematical model along with some constructive and improvement heuristics to solve the parallel blocking flow shop problem (PBFSP) and thus minimize the maximum completion time among lines. The proposed constructive procedures use two approaches that are totally different from those proposed in the literature. These methods are used as initial solution procedures of an iterated local search (ILS) and an iterated greedy algorithm (IGA), both of which are combined with a variable neighborhood search (VNS). The proposed constructive procedure and the improved methods take into account the characteristics of the problem. The computational evaluation demonstrates that both of them –especially the IGA– perform considerably better than those algorithms adapted from the DPFSP literature.Peer ReviewedPostprint (author's final draft