Non-uniform Geometric Set Cover and Scheduling on Multiple Machines

We consider the following general scheduling problem studied recently by Moseley. There are $n$ jobs, all released at time $0$, where job $j$ has size $p_j$ and an associated arbitrary non-decreasing cost function $f_j$ of its completion time. The goal is to find a schedule on $m$ machines with minimum total cost. We give an $O(1)$ approximation for the problem, improving upon the previous $O(\log \log nP)$ bound ($P$ is the maximum to minimum size ratio), and resolving the open question of Moseley. We first note that the scheduling problem can be reduced to a clean geometric set cover problem where points on a line with arbitrary demands, must be covered by a minimum cost collection of given intervals with non-uniform capacity profiles. Unfortunately, current techniques for such problems based on knapsack cover inequalities and low union complexity, completely lose the geometric structure in the non-uniform capacity profiles and incur at least an $\Omega(\log\log P)$ loss. To this end, we consider general covering problems with non-uniform capacities, and give a new method to handle capacities in a way that completely preserves their geometric structure. This allows us to use sophisticated geometric ideas in a black-box way to avoid the $\Omega(\log \log P)$ loss in previous approaches. In addition to the scheduling problem above, we use this approach to obtain $O(1)$ or inverse Ackermann type bounds for several basic capacitated covering problems

Analysis of Dynamic Memory Bandwidth Regulation in Multi-core Real-Time Systems

One of the primary sources of unpredictability in modern multi-core embedded systems is contention over shared memory resources, such as caches, interconnects, and DRAM. Despite significant achievements in the design and analysis of multi-core systems, there is a need for a theoretical framework that can be used to reason on the worst-case behavior of real-time workload when both processors and memory resources are subject to scheduling decisions. In this paper, we focus our attention on dynamic allocation of main memory bandwidth. In particular, we study how to determine the worst-case response time of tasks spanning through a sequence of time intervals, each with a different bandwidth-to-core assignment. We show that the response time computation can be reduced to a maximization problem over assignment of memory requests to different time intervals, and we provide an efficient way to solve such problem. As a case study, we then demonstrate how our proposed analysis can be used to improve the schedulability of Integrated Modular Avionics systems in the presence of memory-intensive workload.Comment: Accepted for publication in the IEEE Real-Time Systems Symposium (RTSS) 2018 conferenc

Truthful Online Scheduling with Commitments

We study online mechanisms for preemptive scheduling with deadlines, with the goal of maximizing the total value of completed jobs. This problem is fundamental to deadline-aware cloud scheduling, but there are strong lower bounds even for the algorithmic problem without incentive constraints. However, these lower bounds can be circumvented under the natural assumption of deadline slackness, i.e., that there is a guaranteed lower bound $s > 1$ on the ratio between a job's size and the time window in which it can be executed. In this paper, we construct a truthful scheduling mechanism with a constant competitive ratio, given slackness $s > 1$. Furthermore, we show that if $s$ is large enough then we can construct a mechanism that also satisfies a commitment property: it can be determined whether or not a job will finish, and the requisite payment if so, well in advance of each job's deadline. This is notable because, in practice, users with strict deadlines may find it unacceptable to discover only very close to their deadline that their job has been rejected

Power Strip Packing of Malleable Demands in Smart Grid

We consider a problem of supplying electricity to a set of $\mathcal{N}$ customers in a smart-grid framework. Each customer requires a certain amount of electrical energy which has to be supplied during the time interval $[0,1]$. We assume that each demand has to be supplied without interruption, with possible duration between $\ell$ and $r$, which are given system parameters ($\ell\le r$). At each moment of time, the power of the grid is the sum of all the consumption rates for the demands being supplied at that moment. Our goal is to find an assignment that minimizes the {\it power peak} - maximal power over $[0,1]$ - while satisfying all the demands. To do this first we find the lower bound of optimal power peak. We show that the problem depends on whether or not the pair $\ell, r$ belongs to a "good" region $\mathcal{G}$. If it does - then an optimal assignment almost perfectly "fills" the rectangle $time \times power = [0,1] \times [0, A]$ with $A$ being the sum of all the energy demands - thus achieving an optimal power peak $A$. Conversely, if $\ell, r$ do not belong to $\mathcal{G}$, we identify the lower bound $\bar{A} >A$ on the optimal value of power peak and introduce a simple linear time algorithm that almost perfectly arranges all the demands in a rectangle $[0, A /\bar{A}] \times [0, \bar{A}]$ and show that it is asymptotically optimal

A Bicriteria Approximation for the Reordering Buffer Problem

In the reordering buffer problem (RBP), a server is asked to process a sequence of requests lying in a metric space. To process a request the server must move to the corresponding point in the metric. The requests can be processed slightly out of order; in particular, the server has a buffer of capacity k which can store up to k requests as it reads in the sequence. The goal is to reorder the requests in such a manner that the buffer constraint is satisfied and the total travel cost of the server is minimized. The RBP arises in many applications that require scheduling with a limited buffer capacity, such as scheduling a disk arm in storage systems, switching colors in paint shops of a car manufacturing plant, and rendering 3D images in computer graphics. We study the offline version of RBP and develop bicriteria approximations. When the underlying metric is a tree, we obtain a solution of cost no more than 9OPT using a buffer of capacity 4k + 1 where OPT is the cost of an optimal solution with buffer capacity k. Constant factor approximations were known previously only for the uniform metric (Avigdor-Elgrabli et al., 2012). Via randomized tree embeddings, this implies an O(log n) approximation to cost and O(1) approximation to buffer size for general metrics. Previously the best known algorithm for arbitrary metrics by Englert et al. (2007) provided an O(log^2 k log n) approximation without violating the buffer constraint.Comment: 13 page

Measuring the Impact of Adversarial Errors on Packet Scheduling Strategies

In this paper we explore the problem of achieving efficient packet transmission over unreliable links with worst case occurrence of errors. In such a setup, even an omniscient offline scheduling strategy cannot achieve stability of the packet queue, nor is it able to use up all the available bandwidth. Hence, an important first step is to identify an appropriate metric for measuring the efficiency of scheduling strategies in such a setting. To this end, we propose a relative throughput metric which corresponds to the long term competitive ratio of the algorithm with respect to the optimal. We then explore the impact of the error detection mechanism and feedback delay on our measure. We compare instantaneous error feedback with deferred error feedback, that requires a faulty packet to be fully received in order to detect the error. We propose algorithms for worst-case adversarial and stochastic packet arrival models, and formally analyze their performance. The relative throughput achieved by these algorithms is shown to be close to optimal by deriving lower bounds on the relative throughput of the algorithms and almost matching upper bounds for any algorithm in the considered settings. Our collection of results demonstrate the potential of using instantaneous feedback to improve the performance of communication systems in adverse environments