A survey of offline algorithms for energy minimization under deadline constraints

Modern computers allow software to adjust power management settings like speed and sleep modes to decrease the power consumption, possibly at the price of a decreased performance. The impact of these techniques mainly depends on the schedule of the tasks. In this article, a survey on underlying theoretical results on power management, as well as offline scheduling algorithms that aim at minimizing the energy consumption under real-time constraints, is given

New Results on Online Resource Minimization

We consider the online resource minimization problem in which jobs with hard deadlines arrive online over time at their release dates. The task is to determine a feasible schedule on a minimum number of machines. We rigorously study this problem and derive various algorithms with small constant competitive ratios for interesting restricted problem variants. As the most important special case, we consider scheduling jobs with agreeable deadlines. We provide the first constant ratio competitive algorithm for the non-preemptive setting, which is of particular interest with regard to the known strong lower bound of n for the general problem. For the preemptive setting, we show that the natural algorithm LLF achieves a constant ratio for agreeable jobs, while for general jobs it has a lower bound of Omega(n^(1/3)). We also give an O(log n)-competitive algorithm for the general preemptive problem, which improves upon the known O(p_max/p_min)-competitive algorithm. Our algorithm maintains a dynamic partition of the job set into loose and tight jobs and schedules each (temporal) subset individually on separate sets of machines. The key is a characterization of how the decrease in the relative laxity of jobs influences the optimum number of machines. To achieve this we derive a compact expression of the optimum value, which might be of independent interest. We complement the general algorithmic result by showing lower bounds that rule out that other known algorithms may yield a similar performance guarantee

On a reduction for a class of resource allocation problems

In the resource allocation problem (RAP), the goal is to divide a given amount of resource over a set of activities while minimizing the cost of this allocation and possibly satisfying constraints on allocations to subsets of the activities. Most solution approaches for the RAP and its extensions allow each activity to have its own cost function. However, in many applications, often the structure of the objective function is the same for each activity and the difference between the cost functions lies in different parameter choices such as, e.g., the multiplicative factors. In this article, we introduce a new class of objective functions that captures the majority of the objectives occurring in studied applications. These objectives are characterized by a shared structure of the cost function depending on two input parameters. We show that, given the two input parameters, there exists a solution to the RAP that is optimal for any choice of the shared structure. As a consequence, this problem reduces to the quadratic RAP, making available the vast amount of solution approaches and algorithms for the latter problem. We show the impact of our reduction result on several applications and, in particular, we improve the best known worst-case complexity bound of two important problems in vessel routing and processor scheduling from $O(n^2)$ to $O(n \log n)$

Models and algorithms for energy-efficient scheduling with immediate start of jobs

We study a scheduling model with speed scaling for machines and the immediate start requirement for jobs. Speed scaling improves the system performance, but incurs the energy cost. The immediate start condition implies that each job should be started exactly at its release time. Such a condition is typical for modern Cloud computing systems with abundant resources. We consider two cost functions, one that represents the quality of service and the other that corresponds to the cost of running. We demonstrate that the basic scheduling model to minimize the aggregated cost function with n jobs is solvable in O(nlogn) time in the single-machine case and in O(n²m) time in the case of m parallel machines. We also address additional features, e.g., the cost of job rejection or the cost of initiating a machine. In the case of a single machine, we present algorithms for minimizing one of the cost functions subject to an upper bound on the value of the other, as well as for finding a Pareto-optimal solution

Modeling and Algorithmic Development for Selected Real-World Optimization Problems with Hard-to-Model Features

Mathematical optimization is a common tool for numerous real-world optimization problems. However, in some application domains there is a scope for improvement of currently used optimization techniques. For example, this is typically the case for applications that contain features which are difficult to model, and applications of interdisciplinary nature where no strong optimization knowledge is available. The goal of this thesis is to demonstrate how to overcome these challenges by considering five problems from two application domains. The first domain that we address is scheduling in Cloud computing systems, in which we investigate three selected problems. First, we study scheduling problems where jobs are required to start immediately when they are submitted to the system. This requirement is ubiquitous in Cloud computing but has not yet been addressed in mathematical scheduling. Our main contributions are (a) providing the formal model, (b) the development of exact and efficient solution algorithms, and (c) proofs of correctness of the algorithms. Second, we investigate the problem of energy-aware scheduling in Cloud data centers. The objective is to assign computing tasks to machines such that the energy required to operate the data center, i.e., the energy required to operate computing devices plus the energy required to cool computing devices, is minimized. Our main contributions are (a) the mathematical model, and (b) the development of efficient heuristics. Third, we address the problem of evaluating scheduling algorithms in a realistic environment. To this end we develop an approach that supports mathematicians to evaluate scheduling algorithms through simulation with realistic instances. Our main contributions are the development of (a) a formal model, and (b) efficient heuristics. The second application domain considered is powerline routing. We are given two points on a geographic area and respective terrain characteristics. The objective is to find a ``good'' route (which depends on the terrain), connecting both points along which a powerline should be built. Within this application domain, we study two selected problems. First, we study a geometric shortest path problem, an abstract and simplified version of the powerline routing problem. We introduce the concept of the k-neighborhood and contribute various analytical results. Second, we investigate the actual powerline routing problem. To this end, we develop algorithms that are built upon the theoretical insights obtained in the previous study. Our main contributions are (a) the development of exact algorithms and efficient heuristics, and (b) a comprehensive evaluation through two real-world case studies. Some parts of the research presented in this thesis have been published in refereed publications [119], [110], [109]