A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths

In the unsplittable flow problem on a path, we are given a capacitated path $P$ and $n$ tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not exceed the capacity of $e$. This is a well-studied problem that has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of $O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any $\epsilon>0$. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a $(2+\epsilon)$-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1, but the presentation has been greatly revised and improved. References have been adde

Integrating Pragmatic Constraints and Behaviors into Real-Time Scheduling Theory

Scheduling theory has been studied and developed extensively in prior research. In some existing scheduling theory results, the focus is primarily on demonstrating interesting theoretical properties, thus these results are not always cognizant of pragmatic constraints. We seek to determine how existing scheduling theory can be improved with respect to pragmatic constraints and behaviors. The goal of this research is to study and design scheduling algorithms for scheduling real-time workload under constraints and behaviors found in real-time systems. Based on our study we derive a scheduling algorithm for partitioning a collection of real-time tasks in a manner that is cognizant of multiple resource constraints. We apply the above scheduling algorithm for partitioning mixed-criticality tasks. In real-time systems the scheduling algorithm must schedule workload such that all timing constraints are met; we verify this using schedulability tests. We describe schedulability tests for each of the scheduling algorithms that we derive. We also propose a new schedulability test for an existing scheduling algorithm that is commonly used in real-time systems research for scheduling tasks with limited-preemptivity. Finally, we propose a scheduling algorithm and schedulability test for scheduling real-time workload on processors that allow dynamic overclocking.Doctor of Philosoph

The multiprocessor real-time scheduling of general task systems

The recent emergence of multicore and related technologies in many commercial systems has increased the prevalence of multiprocessor architectures. Contemporaneously, real-time applications have become more complex and sophisticated in their behavior and interaction. Inevitably, these complex real-time applications will be deployed upon these multiprocessor platforms and require temporal analysis techniques to verify their correctness. However, most prior research in multiprocessor real-time scheduling has addressed the temporal analysis only of Liu and Layland task systems. The goal of this dissertation is to extend real-time scheduling theory for multiprocessor systems by developing temporal analysis techniques for more general task models such as the sporadic task model, the generalized multiframe task model, and the recurring real-time task model. The thesis of this dissertation is: Optimal online multiprocessor real-time scheduling algorithms for sporadic and more general task systems are impossible; however, efficient, online scheduling algorithms and associated feasibility and schedulability tests, with provably bounded deviation from any optimal test, exist. To support our thesis, this dissertation develops feasibility and schedulability tests for various multiprocessor scheduling paradigms. We consider three classes of multiprocessor scheduling based on whether a real-time job may migrate between processors: full-migration, restricted-migration, and partitioned. For all general task systems, we obtain feasibility tests for arbitrary real-time instances under the full-and restricted-migration paradigms. Despite the existence of tests for feasibility, we show that optimal online scheduling of sporadic and more general systems is impossible. Therefore, we focus on scheduling algorithms that have constant-factor approximation ratios in terms of an analysis technique known as resource augmentation. We develop schedulability tests for scheduling algorithms, earliest-deadline-first (edf) and deadline-monotonic (dm), under full-migration and partitioned scheduling paradigms. Feasibility and schedulability tests presented in this dissertation use the workload metrics of demand-based load and maximum job density and have provably bounded deviation from optimal in terms of resource augmentation. We show the demand-based load and maximum job density metrics may be exactly computed in pseudo-polynomial time for general task systems and approximated in polynomial time for sporadic task systems

An Average-Case Analysis for Rate-Monotonic Multiprocessor Real-time Scheduling

We introduce the "First Fit Matching Periods" algorithm for static-priority multiprocessor scheduling of periodic tasks with implicit deadlines and show that it yields asymptotically optimal processor assignments if utilization values are chosen uniformly at random. More precisely we prove that the expected waste is upper bounded by O(n^(3/4) * (log n)^(3/8)). Here the waste denotes the ratio of idle times, cumulated over all processors and n gives the number of tasks. The algorithm can be implemented to run in time O(n log n) and even in the worst case, an asymptotic approximation ratio of 2 is guaranteed. Experiments yield an expected waste proportional to n^0.70, indicating that the above upper bound on the expected waste is almost tight

Algorithms for Scheduling Problems and Integer Programming

The first part of this thesis gives approximation results to scheduling problems. The classical makespan minimization problem on identical parallel machines asks for a distribution of a set of jobs to a set of machines such that the latest job completion time is minimized. For this strongly NP-complete problem we give a new EPTAS algorithm. In fact, it admits a practical implementation which beats the currently best approximation ratio of the MULTIFIT algorithm. A well-studied extension of the problem is the partition of the jobs into classes which impose a class-specific setup time on a machine whenever the processing switches to a job of a different class. For these so-called scheduling problems with batch setup times we present a 1.5-approximation algorithm for each of the three major settings. We achieve similar results for the likewise natural variant of many shared resources scheduling (MSRS) where instead of imposing a setup time each class is identified by a resource which can be occupied by at most one of its jobs at a time. For MSRS we present a 1.5-approximation and two EPTAS results. The second part provides results for fixed-priority uniprocessor real-time scheduling and variants of block-structured integer programming. We give a new approach to compute worst-case response times which admits a polynomial-time algorithm for harmonic periods even in the presence of task release jitters. In more detail, we prove a duality between Response Time Computation (RTC) and the Mixing Set problem. Furthermore, both problems can be expressed as block-structured integer programs which are closely related to simultaneous congruences. However, the setting of the famous Chinese Remainder Theorem is that each congruence has to have a certain remainder. We relax this setting such that the remainder of each congruence may lie in a given interval. We show that the smallest solution to these congruences can be computed in polynomial time if the set of divisors is harmonic

Online Algorithms for Dynamic Resource Allocation Problems

Dynamic resource allocation problems are everywhere. Airlines reserve flight seats for those who purchase flight tickets. Healthcare facilities reserve appointment slots for patients who request them. Freight carriers such as motor carriers, railroad companies, and shipping companies pack containers with loads from specific origins to destinations. We focus on optimizing such allocation problems where resources need to be assigned to customers in real time. These problems are particularly difficult to solve because they depend on random external information that unfolds gradually over time, and the number of potential solutions is overwhelming to search through by conventional methods. In this dissertation, we propose viable allocation algorithms for industrial use, by fully leveraging data and technology to produce gains in efficiency, productivity, and usability of new systems. The first chapter presents a summary of major methodologies used in modeling and algorithm design, and how the methodologies are driven by the size of accessible data. Chapters 2 to 5 present genuine research results of resource allocation problems that are based on Wang and Truong (2017); Wang et al. (2015); Stein et al. (2017); Wang et al. (2016). The algorithms and models cover problems in multiple industries, from a small clinic that aims to better utilize its expensive medical devices, to a technology giant that needs a cost-effective, distributed resource-allocation algorithm in order to maintain the relevance of its advertisements to hundreds of millions of consumers

Approximate feasibility in real-time scheduling: Speeding up in order to meet deadlines

Stougie, L. [Promotor]Marchetti-Spaccamela, A. [Promotor