Scheduling under Linear Constraints

We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job completion time among all feasible choices. This novel problem is motivated by various real-world application scenarios. We discuss the computational complexity and algorithms for various settings of this problem. In particular, we show that if there is only one machine with an arbitrary number of linear constraints, or there is an arbitrary number of machines with no more than two linear constraints, or both the number of machines and the number of linear constraints are fixed constants, then the problem is polynomial-time solvable via solving a series of linear programming problems. If both the number of machines and the number of constraints are inputs of the problem instance, then the problem is NP-Hard. We further propose several approximation algorithms for the latter case.Comment: 21 page

Robust job-sequencing with an uncertain flexible maintenance activity

In this study, the problem of scheduling a set of jobs and one uncertain maintenance activity on a single machine, with the objective of minimizing the makespan is addressed. The maintenance activity has a given duration and must be executed within a given time window. Furthermore, duration and time window of the maintenance are uncertain, and can take different values which can be described by different scenarios. The problem is to determine a job sequence which performs well, in terms of makespan, independently on the possible variation of the data concerning the maintenance. A robust scheduling approach is used for the problem, in which four different measures of robustness are considered, namely, maximum absolute regret, maximum relative regret, worst-case scenario, and ordered weighted averaging. Complexity and approximation results are presented. In particular, we show that, for all the four robustness criteria, the problem is strongly NP-hard. A number of special cases are explored, and an exact pseudopolynomial algorithm based on dynamic programming is devised when the number of scenarios is fixed. Two Mixed Integer Programming (MIP) models are also presented for the general problem. Several computational experiments have been conducted to evaluate the efficiency and effectiveness of the MIP models and of the dynamic programming approach

Database query optimisation based on measures of regret

The query optimiser in a database management system (DBMS) is responsible for �nding a good order in which to execute the operators in a given query. However, in practice the query optimiser does not usually guarantee to �nd the best plan. This is often due to the non-availability of precise statistical data or inaccurate assumptions made by the optimiser. In this thesis we propose a robust approach to logical query optimisation that takes into account the unreliability in database statistics during the optimisation process. In particular, we study the ordering problem for selection operators and for join operators, where selectivities are modelled as intervals rather than exact values. As a measure of optimality, we use a concept from decision theory called minmax regret optimisation (MRO). When using interval selectivities, the decision problem for selection operator ordering turns out to be NP-hard. After investigating properties of the problem and identifying special cases which can be solved in polynomial time, we develop a novel heuristic for solving the general selection ordering problem in polynomial time. Experimental evaluation of the heuristic using synthetic data, the Star Schema Benchmark and real-world data sets shows that it outperforms other heuristics (which take an optimistic, pessimistic or midpoint approach) and also produces plans whose regret is on average very close to optimal. The general join ordering problem is known to be NP-hard, even for exact selectivities. So, for interval selectivities, we restrict our investigation to sets of join operators which form a chain and to plans that correspond to left-deep join trees. We investigate properties of the problem and use these, along with ideas from the selection ordering heuristic and other algorithms in the literature, to develop a polynomial-time heuristic tailored for the join ordering problem. Experimental evaluation of the heuristic shows that, once again, it performs better than the optimistic, pessimistic and midpoint heuristics. In addition, the results show that the heuristic produces plans whose regret is on average even closer to the optimal than for selection ordering

Towards Understanding Uncertainty in Cloud Computing Resource Provisioning

In spite of extensive research of uncertainty issues in different fields ranging from computational biology to decision making in economics, a study of uncertainty for cloud computing systems is limited. Most of works examine uncertainty phenomena in users’ perceptions of the qualities, intentions and actions of cloud providers, privacy, security and availability. But the role of uncertainty in the resource and service provisioning, programming models, etc. have not yet been adequately addressed in the scientific literature. There are numerous types of uncertainties associated with cloud computing, and one should to account for aspects of uncertainty in assessing the efficient service provisioning. In this paper, we tackle the research question: what is the role of uncertainty in cloud computing service and resource provisioning? We review main sources of uncertainty, fundamental approaches for scheduling under uncertainty such as reactive, stochastic, fuzzy, robust, etc. We also discuss potentials of these approaches for scheduling cloud computing activities under uncertainty, and address methods for mitigating job execution time uncertainty in the resource provisioning.Peer ReviewedPostprint (published version

Scheduling Models with Additional Features: Synchronization, Pliability and Resiliency

In this thesis we study three new extensions of scheduling models with both practical and theoretical relevance, namely synchronization, pliability and resiliency. Synchronization has previously been studied for flow shop scheduling and we now apply the concept to open shop models for the first time. Here, as opposed to the traditional models, operations that are processed together all have to be started at the same time. Operations that are completed are not removed from the machines until the longest operation in their group is finished. Pliability is a new approach to model flexibility in flow shops and open shops. In scheduling with pliability, parts of the processing load of the jobs can be re-distributed between the machines in order to achieve better schedules. This is applicable, for example, if the machines represent cross-trained workers. Resiliency is a new measure for the quality of a given solution if the input data are uncertain. A resilient solution remains better than some given bound, even if the original input data are changed. The more we can perturb the input data without the solution losing too much quality, the more resilient the solution is. We also consider the assignment problem, as it is the traditional combinatorial optimization problem underlying many scheduling problems. Particularly, we study a version of the assignment problem with a special cost structure derived from the synchronous open shop model and obtain new structural and complexity results. Furthermore we study resiliency for the assignment problem. The main focus of this thesis is the study of structural properties, algorithm development and complexity. For synchronous open shop we show that for a fixed number of machines the makespan can be minimized in polynomial time. All other traditional scheduling objectives are at least as hard to optimize as in the traditional open shop model. Starting out research in pliability we focus on the most general case of the model as well as two relevant special cases. We deliver a fairly complete complexity study for all three versions of the model. Finally, for resiliency, we investigate two different questions: `how to compute the resiliency of a given solution?' and `how to find a most resilient solution?'. We focus on the assignment problem and single machine scheduling to minimize the total sum of completion times and present a number of positive results for both questions. The main goal is to make a case that the concept deserves further study