In this paper we initiate the study of minimizing power consumption in the broadcast scheduling model. In this setting there is a wireless transmitter. Over time requests arrive at the transmitter for pages of information. Multiple requests may be for the same page. When a page is transmitted, all requests for that page receive the transmission simulteneously. The speed the transmitter sends data at can be dynamically scaled to conserve energy. We consider the problem of minimizing flow time plus energy, the most popular scheduling metric considered in the standard scheduling model when the scheduler is energy aware. We will assume that the power consumed is modeled by an arbitrary convex function. For this problem there is a $Ω(n)$ lower bound. Due to the lower bound, we consider the resource augmentation model of Gupta \etal \cite{GuptaKP10}. Using resource augmentation, we give a scalable algorithm. Our result also gives a scalable non-clairvoyant algorithm for minimizing weighted flow time plus energy in the standard scheduling model.
We consider online algorithms for broadcast scheduling. In the pull-based broadcast model there are $n$ unit-sized pages of information at a server and requests arrive online for pages. When the server transmits a page $p$, all outstanding requests for that page are satisfied. The longest-wait-first} (LWF) algorithm is a natural algorithm that has been shown to have good empirical performance. In this paper we make two main contributions to the analysis of LWF and broadcast scheduling. \begin{itemize} \item We give an intuitive and easy to understand analysis of LWF which shows that it is $O(1…
We consider the pull-based broadcast scheduling model. In this model, there are n unit-sized pages of information available at the server. Requests arrive over time at the server asking for a specific page. When the server transmits a page, all outstanding requests for the page are simultaneously satisfied, and this is what distinguishes broadcast scheduling from the standard scheduling setting where each job must be processed separately by the server. Broadcast scheduling has received a considerable amount of attention due to the algorithmic challenges that it gives in addition to its applica…
The broadcast scheduling problem asks how a multihop network of broadcast transceivers operating on a shared medium may share the medium in such a way that communication over the entire network is possible. This can be naturally modeled as a graph coloring problem via distance-2 coloring (L(1,1)-labeling, strict scheduling). This coloring is difficult to compute and may require a number of colors quadratic in the graph degree. This paper introduces pseudo-scheduling, a relaxation of distance-2 coloring. Centralized and decentralized algorithms that compute pseudo-schedules with colors linear i…
We consider online algorithms for pull-based broadcast scheduling. In this setting there are n pages of information at a server and requests for pages arrive online. When the server serves (broadcasts) a page p, all outstanding requests for that page are satisfied. We study two related metrics, namely maximum response time (waiting time) and maximum delay-factor and their weighted versions. We obtain the following results in the worst-case online competitive model. - We show that FIFO (first-in first-out) is 2-competitive even when the page sizes are different. Previously this was known only f…
In this paper two scheduling models are addressed. First is the standard model (unicast) where requests (or jobs) are independent. The other is the broadcast model where broadcasting a page can satisfy multiple outstanding requests for that page. We consider online scheduling of requests when they have deadlines. Unlike previous models, which mainly consider the objective of maximizing throughput while respecting deadlines, here we focus on scheduling all the given requests with the goal of minimizing the maximum {\em delay factor}.We prove strong lower bounds on the achievable competitive rat…
This report contains revision and extension of some results about RBO [arXiv:1108.5095]. RBO is a simple and efficient broadcast scheduling of $n = 2^k$ uniform frames for battery powered radio receivers. Each frame contains a key from some arbitrary linearly ordered universe. The broadcast cycle -- a sequence of frames sorted by the keys and permuted by $k$-bit reversal -- is transmitted in a round robin fashion by the broadcaster. At arbitrary time during the transmission, the receiver may start a simple protocol that reports to him all the frames with the keys that are contained in a specif…
Given n jobs with release dates, deadlines and processing times we consider the problem of scheduling them on m parallel machines so as to minimize the total energy consumed. Machines can enter a sleep state and they consume no energy in this state. Each machine requires Q units of energy to awaken from the sleep state and in its active state the machine can process jobs and consumes a unit of energy per unit time. We allow for preemption and migration of jobs and provide the first constant approximation algorithm for this problem.
We consider the class of single machine scheduling problems with the objective to minimize the weighted number of late jobs, under the assumption that completion due-dates are not known precisely at the time when decision-maker must provide a schedule. It is assumed that only the intervals to which the due-dates belong are known. The concept of maximum regret is used to define robust solution. A polynomial time algorithm is given for the case when weights of jobs are all equal. A mixed-integer linear programming formulation is provided for the general case, and computational experiments are re…
We study the problem of preemptive scheduling of n equal-length jobs with given release times on m identical parallel machines. The objective is to minimize the average flow time. Recently, Brucker and Kravchenko proved that the optimal schedule can be computed in polynomial time by solving a linear program with O(n^3) variables and constraints, followed by some substantial post-processing (where n is the number of jobs.) In this note we describe a simple linear program with only O(mn) variables and constraints. Our linear program produces directly the optimal schedule and does not require any…
We study the online scheduling problem where the machines need to be calibrated before processing any jobs. To calibrate a machine, it will take $λ$ time steps as the activation time, and then the machine will remain calibrated status for $T$ time steps. The job can only be processed by the machine that is in calibrated status. Given a set of jobs arriving online, each of the jobs is characterized by a release time, a processing time, and a deadline. We assume that there is an infinite number of machines for usage. The objective is to minimize the total number of calibrations while feasibly sc…
We study approximation algorithms for scheduling problems with the objective of minimizing total weighted completion time, under identical and related machine models with job precedence constraints. We give algorithms that improve upon many previous 15 to 20-year-old state-of-art results. A major theme in these results is the use of time-indexed linear programming relaxations. These are natural relaxations for their respective problems, but surprisingly are not studied in the literature. We also consider the scheduling problem of minimizing total weighted completion time on unrelated machines.…
In this paper, we introduce a new online scheduling framework for minimizing total weighted completion time in a general setting. The framework is inspired by the work of Hall et al. [Mathematics of Operations Research, Vol 22(3):513-544, 1997] and Garg et al. [Proc. of Foundations of Software Technology and Theoretical Computer Science, pp. 96-107, 2007], who show how to convert an offline approximation to an online scheme. Our framework uses two offline approximation algorithms (one for the simpler problem of scheduling without release times, and another for the minimum unscheduled weight pr…
We introduce and study a general scheduling problem that we term the Packing Scheduling problem. In this problem, jobs can have different arrival times and sizes; a scheduler can process job $j$ at rate $x_j$, subject to arbitrary packing constraints over the set of rates ($\vec{x}$) of the outstanding jobs. The PSP framework captures a variety of scheduling problems, including the classical problems of unrelated machines scheduling, broadcast scheduling, and scheduling jobs of different parallelizability. It also captures scheduling constraints arising in diverse modern environments ranging f…
Scheduling a set of jobs over a collection of machines is a fundamental problem that needs to be solved millions of times a day in various computing platforms: in operating systems, in large data clusters, and in data centers. Along with makespan, flow-time, which measures the length of time a job spends in a system before it completes, is arguably the most important metric to measure the performance of a scheduling algorithm. In recent years, there has been a remarkable progress in understanding flow-time based objective functions in diverse settings such as unrelated machines scheduling, bro…
Efficient Approximation Algorithms for Scheduling Coflows with Precedence Constraints in Identical Parallel Networks to Minimize Weighted Completion Time
The $1|B,r_j|\sum w_jU_j$ scheduling problem takes as input a batch setup time $Δ$ and a set of $n$ jobs, each having a processing time, a release date, a weight, and a due date; the task is to find a sequence of batches that minimizes the weighted number of tardy jobs. This problem was introduced by Hochbaum and Landy in 1994; as a wide generalization of {\sc Knapsack}, it is $\mathsf{NP}$-hard. In this work we provide a multivariate complexity analysis of the $1|B,r_j|\sum w_jU_j$ problem with respect to several natural parameters. That is, we establish a thorough classification into fixed-p…
Consider a problem in which $n$ jobs that are classified into $k$ types arrive over time at their release times and are to be scheduled on a single machine so as to minimize the maximum flow time. The machine requires a setup taking $s$ time units whenever it switches from processing jobs of one type to jobs of a different type. We consider the problem as an online problem where each job is only known to the scheduler as soon as it arrives and where the processing time of a job only becomes known upon its completion (non-clairvoyance). We are interested in the potential of simple "greedy-like"…
In this paper, we consider the online problem of scheduling independent jobs \emph{non-preemptively} so as to minimize the weighted flow-time on a set of unrelated machines. There has been a considerable amount of work on this problem in the preemptive setting where several competitive algorithms are known in the classical competitive model. %Using the speed augmentation model, Anand et al. showed that the greedy algorithm is $O\left(\frac{1}ε\right)$-competitive in the preemptive setting. In the non-preemptive setting, Lucarelli et al. showed that there exists a strong lower bound for minimiz…
In classical scheduling problems, we are given jobs and machines, and have to schedule all the jobs to minimize some objective function. What if each job has a specified profit, and we are no longer required to process all jobs -- we can schedule any subset of jobs whose total profit is at least a (hard) target profit requirement, while still approximately minimizing the objective function? We refer to this class of problems as scheduling with outliers. This model was initiated by Charikar and Khuller (SODA'06) on the minimum max-response time in broadcast scheduling. We consider three other w…
We consider online scheduling to minimize weighted completion time on related machines, where each job consists of several tasks that can be concurrently executed. A job gets completed when all its component tasks finish. We obtain an $O(K^3 \log^2 K)$-competitive algorithm in the non-clairvoyant setting, where $K$ denotes the number of distinct machine speeds. The analysis is based on dual-fitting on a precedence-constrained LP relaxation that may be of independent interest.
The active-time scheduling problem considers the problem of scheduling preemptible jobs with windows (release times and deadlines) on a parallel machine that can schedule up to $g$ jobs during each timestep. The goal in the active-time problem is to minimize the number of active steps, i.e., timesteps in which at least one job is scheduled. In this way, the active time models parallel scheduling when there is a fixed cost for turning the machine on at each discrete step. This paper presents a 9/5-approximation algorithm for a special case of the active-time scheduling problem in which job wind…
The data broadcast problem is to find a schedule for broadcasting a given set of messages over multiple channels. The goal is to minimize the cost of the broadcast plus the expected response time to clients who periodically and probabilistically tune in to wait for particular messages. The problem models disseminating data to clients in asymmetric communication environments, where there is a much larger capacity from the information source to the clients than in the reverse direction. Examples include satellites, cable TV, internet broadcast, and mobile phones. Such environments favor the ``pu…
This paper proposes a new formulation for the school bus scheduling problem (SBSP) which optimizes school start times and bus operation times to minimize transportation cost. Our goal is to minimize the number of buses to serve all bus routes such that each route arrives in a time window before school starts. We present a new time-indexed integer linear programming (ILP) formulation for this problem. Based on a strengthened version of the linear relaxation of the ILP, we develop a dependent randomized rounding algorithm that yields near-optimal solutions for large-scale problem instances. We a…
Broadcasting is known to be an efficient means of disseminating data in wireless communication environments (such as Satellite, mobile phone networks,...). It has been recently observed that the average service time of broadcast systems can be considerably improved by taking into consideration existing correlations between requests. We study a pull-based data broadcast system where users request possibly overlapping sets of items; a request is served when all its requested items are downloaded. We aim at minimizing the average user perceived latency, i.e. the average flow time of the requests.…
In many professional sports leagues, teams from opposing leagues/conferences compete against one another, playing inter-league games. This is an example of a bipartite tournament. In this paper, we consider the problem of reducing the total travel distance of bipartite tournaments, by analyzing inter-league scheduling from the perspective of discrete optimization. This research has natural applications to sports scheduling, especially for leagues such as the National Basketball Association (NBA) where teams must travel long distances across North America to play all their games, thus consuming…
We consider the problem of scheduling a set of $n$ tasks on $m$ processors under precedence, communication, and global system energy constraints to minimize makespan. We extend existing scheduling models to account for energy usage and give convex programming algorithms that yield essentially the same results as existing algorithms that do not consider energy, while adhering to a strict energy bound.
We consider the problem of throughput-optimal broadcast- ing in time-varying wireless networks, whose underlying topology is restricted to Directed Acyclic Graphs (DAG). Previous broadcast algorithms route packets along spanning trees. In large networks with time-varying connectivities, these trees are difficult to compute and maintain. In this paper we propose a new online throughput-optimal broadcast algorithm which makes packet-by-packet scheduling and routing decisions, obviating the need for maintaining any global topological structures, such as spanning-trees. Our algorithm relies on sys…
In the paper "The complexity of mean flow time scheduling problems with release times", by Baptiste, Brucker, Chrobak, Dürr, Kravchenko and Sourd, the authors claimed to prove strong NP-hardness of the scheduling problem $P|pmtn,r_j|\sum C_j$, namely multiprocessor preemptive scheduling where the objective is to minimize the mean flow time. We point out a serious error in their proof and give a new proof of strong NP-hardness for this problem.
In this paper we study a proportionate flow shop of batching machines with release dates and a fixed number $m \geq 2$ of machines. The scheduling problem has so far barely received any attention in the literature, but recently its importance has increased significantly, due to applications in the industrial scaling of modern bio-medicine production processes. We show that for any fixed number of machines, the makespan and the sum of completion times can be minimized in polynomial time. Furthermore, we show that the obtained algorithm can also be used to minimize the weighted total completion …
We consider the problem of energy-efficient transmission in delay constrained cooperative multihop wireless networks. The combinatorial nature of cooperative multihop schemes makes it difficult to design efficient polynomial-time algorithms for deciding which nodes should take part in cooperation, and when and with what power they should transmit. In this work, we tackle this problem in memoryless networks with or without delay constraints, i.e., quality of service guarantee. We analyze a wide class of setups, including unicast, multicast, and broadcast, and two main cooperative approaches, na…
Applications in data-parallel computing typically consist of multiple stages. In each stage, a set of intermediate parallel data flows (Coflow) is produced and transferred between servers to enable starting of next stage. While there has been much research on scheduling isolated coflows, the dependency between coflows in multi-stage jobs has been largely ignored. In this paper, we consider scheduling coflows of multi-stage jobs represented by general DAGs (Directed Acyclic Graphs) in a shared data center network, so as to minimize the total weighted completion time of jobs. This problem is sig…
The problem of online packet scheduling to minimize the required conventional grid energy for transmitting a fixed number of packets given a common deadline is considered. The total number of packets arriving within the deadline is known, but the packet arrival times are unknown, and can be arbitrary. The proposed algorithm tries to finish the transmission of each packet assuming all future packets are going to arrive at equal time intervals within the left-over time. The proposed online algorithm is shown to have competitive ratio that is logarithmic in the number of packet arrivals. The hybr…
Let $n = 2^k$ be the length of the broadcast cycle of the RBO broadcast scheduling protocol (see [arXiv:1108.5095] and [arXiv:1201.3318]). Let $lb$ and $ub$ be the variables of the RBO receiver as defined in [ arXiv:1201.3318 ]. We show that the number of changes of $lb$ (the "left-side energy") is not greater than $k + 1$. We also show that the number of changes of $rb$ (the "right-side energy") is not greater than $k + 2$. Thus the "extra energy" (defined in [arXiv:1201.3318]) is bounded by $2 k + 3$. This updates the previous bound from [arXiv:1201.3318], which was $4 k + 2$.
