We study the problem of computing a preemptive schedule of equal-length jobs with given release times, deadlines and weights. Our goal is to maximize the weighted throughput, which is the total weight of completed jobs. In Graham's notation this problem is described as (1 | r_j;p_j=p;pmtn | sum w_j U_j). We provide an O(n^4)-time algorithm for this problem, improving the previous bound of O(n^{10}) by Baptiste.
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…
When a computer system schedules jobs there is typically a significant cost associated with preempting a job during execution. This cost can be from the expensive task of saving the memory's state and loading data into and out of memory. It is desirable to schedule jobs non-preemptively to avoid the costs of preemption. There is a need for non-preemptive system schedulers on desktops, servers and data centers. Despite this need, there is a gap between theory and practice. Indeed, few non-preemptive \emph{online} schedulers are known to have strong foundational guarantees. This gap is likely du…
We investigate the problem of scheduling the maintenance of edges in a network, motivated by the goal of minimizing outages in transportation or telecommunication networks. We focus on maintaining connectivity between two nodes over time; for the special case of path networks, this is related to the problem of minimizing the busy time of machines. We show that the problem can be solved in polynomial time in arbitrary networks if preemption is allowed. If preemption is restricted to integral time points, the problem is NP-hard and in the non-preemptive case we give strong non-approximability re…
We consider the online problem of minimizing weighted flow-time on unrelated machines. Although much is known about this problem in the resource-augmentation setting, these results assume that jobs can be preempted. We give the first constant-competitive algorithm for the non-preemptive setting in the rejection model. In this rejection model, we are allowed to reject an $\varepsilon$-fraction of the total weight of jobs, and compare the resulting flow-time to that of the offline optimum which is required to schedule all jobs. This is arguably the weakest assumption in which such a result is kn…
We are given a set of jobs, each one specified by its release date, its deadline and its processing volume (work), and a single (or a set of) speed-scalable processor(s). We adopt the standard model in speed-scaling in which if a processor runs at speed s then the energy consumption is s^α per time unit, where α>1. Our goal is to find a schedule respecting the release dates and the deadlines of the jobs so that the total energy consumption is minimized. While most previous works have studied the preemptive case of the problem, where a job may be interrupted and resumed later, we focus on the n…
Consider the problem in which n jobs that are classified into k types are to be scheduled on m identical machines without preemption. A machine requires a proper setup taking s time units before processing jobs of a given type. The objective is to minimize the makespan of the resulting schedule. We design and analyze an approximation algorithm that runs in time polynomial in n, m and k and computes a solution with an approximation factor that can be made arbitrarily close to 3/2.
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…
We consider a basic problem of preemptive scheduling of $n$ non-simultaneously released jobs on a group of $m$ unrelated parallel machines so as to minimize maximum job completion time, the makespan. In the scheduling literature, the problem is commonly considered to be solvable in polynomial time by linear programming (LP) techniques proposed in Lawler and Labetoulle \cite{ll78}. The authors in \cite{ll78} give a LP formulation of the version with simultaneously released jobs and show how an optimal solution to this LP can be used to construct an optimal schedule to the latter problem. As the…
We are given a set of $n$ jobs and a single processor that can vary its speed dynamically. Each job $J_j$ is characterized by its processing requirement (work) $p_j$, its release date $r_j$ and its deadline $d_j$. We are also given a budget of energy $E$ and we study the scheduling problem of maximizing the throughput (i.e. the number of jobs which are completed on time). We propose a dynamic programming algorithm that solves the preemptive case of the problem, i.e. when the execution of the jobs may be interrupted and resumed later, in pseudo-polynomial time. Our algorithm can be adapted for …
Motivated by the problem to approximate all feasible schedules by one schedule in a given scheduling environment, we introduce in this paper the concepts of strong simultaneous approximation ratio (SAR) and weak simultaneous approximation ratio (WAR). Then we study the two parameters under various scheduling environments, such as, non-preemptive, preemptive or fractional scheduling on identical, related or unrelated machines.
We study the problem of non-preemptively scheduling $n$ jobs, each job $j$ with a release time $t_j$, a deadline $d_j$, and a processing time $p_j$, on $m$ parallel identical machines. Cieliebak et al. (2004) considered the two constraints $|d_j-t_j|\leq λp_j$ and $|d_j-t_j|\leq p_j +σ$ and showed the problem to be NP-hard for any $λ>1$ and for any $σ\geq 2$. We complement their results by parameterized complexity studies: we show that, for any $λ>1$, the problem remains weakly NP-hard even for $m=2$ and strongly W[1]-hard parameterized by $m$. We present a pseudo-polynomial-time algorithm for…
We study a scheduling problem arising in demand response management in smart grid. Consumers send in power requests with a flexible feasible time interval during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. Previous work has studied cases where jobs have unit power requirement and unit duration. We extend the study to arbitrary power …
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 str…
We propose a Boolean Linear Programming model for the preemptive single machine scheduling problem with equal processing times, arbitrary release dates and weights(priorities) minimizing the total weighted completion time. Almost always an optimal solution of the Linear Programming relaxation is integral and can be straightforwardly converted into an optimal schedule. To deal with the fractional solutions we present two heuristics. Very often our heuristics find solutions with objective function values equal to the lower bound found by the Linear Programming relaxation. For the cases when uppe…
We revisit a classical scheduling model to incorporate modern trends in data center networks and cloud services. Addressing some key challenges in the allocation of shared resources to user requests (jobs) in such settings, we consider the following variants of the classic {\em resource allocation problem} (\textsf{RAP}). The input to our problems is a set $J$ of jobs and a set $M$ of homogeneous hosts, each has an available amount of some resource. A job is associated with a release time, a due date, a weight, and a given length, as well as its resource requirement. A \emph{feasible} schedule…
We propose a unifying framework based on configuration linear programs and randomized rounding, for different energy optimization problems in the dynamic speed-scaling setting. We apply our framework to various scheduling and routing problems in heterogeneous computing and networking environments. We first consider the energy minimization problem of scheduling a set of jobs on a set of parallel speed scalable processors in a fully heterogeneous setting. For both the preemptive-non-migratory and the preemptive-migratory variants, our approach allows us to obtain solutions of almost the same qua…
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.
We introduce the following elementary scheduling problem. We are given a collection of n jobs, where each job has an integer length as well as a set Ti of time intervals in which it can be feasibly scheduled. Given a parameter B, the processor can schedule up to B jobs at a timeslot t so long as it is "active" at t. The goal is to schedule all the jobs in the fewest number of active timeslots. The machine consumes a fixed amount of energy per active timeslot, regardless of the number of jobs scheduled in that slot (as long as the number of jobs is non-zero). In other words, subject to all unit…
Assigning jobs onto identical machines with the objective to minimize the maximal load is one of the most basic problems in combinatorial optimization. Motivated by product planing and data placement, we study a natural extension called Class Constrainted Scheduling (CCS). In this problem, each job additionally belongs to a class and each machine can only schedule jobs from at most $c$ different classes. Even though this problem is closely related to the Class Constraint Bin Packing, the Class Constraint Knapsack and the Cardinality Constraint variants, CCS lacks results regarding approximatio…
In this paper, we consider the slack due-window assignment model and study a single machine scheduling problem of linear time-dependent deteriorating jobs and a deteriorating maintenance activity. The cost for each job consists of four components: earliness, tardiness, window location and window size. The objective is to schedule the jobs and to assign the maintenance activity and due-windows such that the total cost among all the jobs is minimized. A polynomial-time algorithm with the running time not exceeding $O(n^2logn)$ to give a solution to this problem is introduced, where $n$ is the nu…
We study the problem of scheduling equal-length jobs with release times and deadlines, where the objective is to maximize the number of completed jobs. Preemptions are not allowed. In Graham's notation, the problem is described as 1|r_j;p_j=p|\sum U_j. We give the following results: (1) We show that the often cited algorithm by Carlier from 1981 is not correct. (2) We give an algorithm for this problem with running time O(n^5).
We consider the problem of scheduling jobs with equal lengths on uniform parallel batch machines with non-identical capacities where each job can only be processed on a specified subset of machines called its processing set. For the case of equal release times, we give efficient exact algorithms for various objective functions. For the case of unequal release times, we give efficient exact algorithms for minimizing makespan.
We investigate the scheduling of $n$ jobs divided into $c$ classes on $m$ identical parallel machines. For every class there is a setup time which is required whenever a machine switches from the processing of one class to another class. The objective is to find a schedule that minimizes the makespan. We give near-linear approximation algorithms for the following problem variants: the non-preemptive context where jobs may not be preempted, the preemptive context where jobs may be preempted but not parallelized, as well as the splittable context where jobs may be preempted and parallelized. We …
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…
Motivated by the current research in data centers and cloud computing, we study the problem of scheduling a set of two-stage jobs on multiple two-stage flowshops. A new formulation for configurations of such scheduling is proposed, which leads directly to improvements to the complexity of scheduling algorithms for the problem. Motivated by the observation that the costs of the two stages can be significantly different, we present deeper study on the structures of the problem that leads to a new approach to designing scheduling algorithms for the problem. With more thorough analysis, we show th…
Co-flows model a modern scheduling setting that is commonly found in a variety of applications in distributed and cloud computing. A stochastic co-flow task contains a set of parallel flows with randomly distributed sizes. Further, many applications require non-preemptive scheduling of co-flow tasks. This paper gives an approximation algorithm for stochastic non-preemptive co-flow scheduling. The proposed approach uses a time-indexed linear relaxation, and uses its solution to come up with a feasible schedule. This algorithm is shown to achieve a competitive ratio of $(2\log{m}+1)(1+\sqrt{m}Δ)…
We study the online preemptive scheduling of intervals and jobs (with restarts). Each interval or job has an arrival time, a deadline, a length and a weight. The objective is to maximize the total weight of completed intervals or jobs. While the deterministic case for intervals was settled a long time ago, the randomized case remains open. In this paper we first give a 2-competitive randomized algorithm for the case of equal length intervals. The algorithm is barely random in the sense that it randomly chooses between two deterministic algorithms at the beginning and then sticks with it therea…
We study the fundamental problem of scheduling bidirectional traffic along a path composed of multiple segments. The main feature of the problem is that jobs traveling in the same direction can be scheduled in quick succession on a segment, while jobs in opposing directions cannot cross a segment at the same time. We show that this tradeoff makes the problem significantly harder than the related flow shop problem, by proving that it is NP-hard even for identical jobs. We complement this result with a PTAS for a single segment and non-identical jobs. If we allow some pairs of jobs traveling in …
We consider an online scheduling problem, motivated by the issues present at the joints of networks using ATM and TCP/IP. Namely, IP packets have to broken down to small ATM cells and sent out before their deadlines, but cells corresponding to different packets can be interwoven. More formally, we consider the online scheduling problem with preemptions, where each job j is revealed at release time r_j, has processing time p_j, deadline d_j and weight w_j. A preempted job can be resumed at any time. The goal is to maximize the total weight of all jobs completed on time. Our main result are as f…
In the SCHED problem we are given a set of n jobs, together with their processing times and precedence constraints. The task is to order the jobs so that their total completion time is minimized. SCHED is a special case of the Traveling Repairman Problem with precedences. A natural dynamic programming algorithm solves both these problems in 2^n n^O(1) time, and whether there exists an algorithms solving SCHED in O(c^n) time for some constant c < 2 was an open problem posted in 2004 by Woeginger. In this paper we answer this question positively.
Machine scheduling problems involving conflict jobs can be seen as a constrained version of the classical scheduling problem, in which some jobs are conflict in the sense that they cannot be proceeded simultaneously on different machines. This conflict constraint naturally arises in several practical applications and has recently received considerable attentions in the research community. In fact, the problem is typically NP-hard (even for approximation) and most of algorithmic results achieved so far have heavily relied on special structures of the underlying graph used to model the conflict-…
In this paper, we study an NP-hard problem of a single machine scheduling minimizing the makespan, where the mixed-critical tasks with an uncertain processing time are scheduled. We show the derivation of F-shaped tasks from the probability distribution function of the processing time, then we study the structure of problems with two and three criticality levels for which we propose efficient exact algorithms and we present computational experiments for instances with up to 200 tasks. Moreover, we show that the considered problem is approximable within a constant multiplicative factor.
We consider online preemptive scheduling of jobs arriving one by one, to be assigned to two identical machines, with the goal of makespan minimization. We study the effect of selecting the best solution out of two independent solutions constructed in parallel in an online fashion. Two cases are analyzed, where one case is purely online, and in the other one jobs are presented sorted by non-increasing sizes. We show that using two solutions rather than one improves the performance significantly, but that an optimal solution cannot be obtained for any constant number of solutions constructed in …
A moldable job is a job that can be executed on an arbitrary number of processors, and whose processing time depends on the number of processors allotted to it. A moldable job is monotone if its work doesn't decrease for an increasing number of allotted processors. We consider the problem of scheduling monotone moldable jobs to minimize the makespan. We argue that for certain compact input encodings a polynomial algorithm has a running time polynomial in n and log(m), where n is the number of jobs and m is the number of machines. We describe how monotony of jobs can be used to counteract the i…
We consider non-preemptive scheduling of MapReduce jobs with multiple tasks in the practical scenario where each job requires several map-reduce rounds. We seek to minimize the average weighted completion time and consider scheduling on identical and unrelated parallel processors. For identical processors, we present LP-based O(1)-approximation algorithms. For unrelated processors, the approximation ratio naturally depends on the maximum number of rounds of any job. Since the number of rounds per job in typical MapReduce algorithms is a small constant, our scheduling algorithms achieve a small…
We analyze the problem of job scheduling with preempting on weighted jobs that can have either linear or exponential penalties. We review relevant literature on the problem and create and describe a few online algorithms that perform competitively with the optimal scheduler. We first describe a na{\" i}ve algorithm, which yields a high competitive ratio ($Ω(\frac{M}{s_{\min}})$) with the optimal, then provide an algorithm that yields a lower competitive ratio ($4\sqrt{\frac{M}{s_{\min}}} + n\log{\frac{Mn}{s_{\min}}}$). Finally, we make a minor modification to our algorithm to yield an algorith…
Scheduling jobs with precedence constraints on a set of identical machines to minimize the total processing time (makespan) is a fundamental problem in combinatorial optimization. In practical settings such as cloud computing, jobs are often malleable, i.e., can be processed on multiple machines simultaneously. The instantaneous processing rate of a job is a non-decreasing function of the number of machines assigned to it (we call it the processing function). Previous research has focused on practically relevant concave processing functions, which obey the law of diminishing utility and genera…
In the paper we consider the problem of scheduling $n$ identical jobs on 4 uniform machines with speeds $s_1 \geq s_2 \geq s_3 \geq s_4,$ respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree $Δ$, where two incompatible jobs cannot be processed on the same machine. We show that the problem is NP-hard even if $s_1=s_2=s_3$. If, however, $Δ\leq 4$ and $s_1 \geq 12 s_2$, $s_2=s_3=s_4$, then the problem can be solved to optimality in time $O(n^{1…
By executing jobs serially rather than in parallel, size-based scheduling policies can shorten time needed to complete jobs; however, major obstacles to their applicability are fairness guarantees and the fact that job sizes are rarely known exactly a-priori. Here, we introduce the Pri family of size-based scheduling policies; Pri simulates any reference scheduler and executes jobs in the order of their simulated completion: we show that these schedulers give strong fairness guarantees, since no job completes later in Pri than in the reference policy. In addition, we introduce PSBS, a practica…
The paper considers scheduling on parallel machines under the constraint that some pairs of jobs cannot be processed concurrently. Each job has an associated weight, and all jobs have the same deadline. The objective is to maximise the total weight of on-time jobs. The problem is known to be strongly NP-hard in general. A polynomial-time algorithm for scheduling unit execution time jobs on two machines is proposed. The performance of a broad family of approximation algorithms for scheduling unit execution time jobs on more than two machines is analysed. For the case of arbitrary job processing…
We consider a natural generalization of scheduling $n$ jobs on $m$ parallel machines so as to minimize the makespan. In our extension the set of jobs is partitioned into several classes and a machine requires a setup whenever it switches from processing jobs of one class to jobs of a different class. During such a setup, a machine cannot process jobs and the duration of a setup may depend on the machine as well as the class of the job to be processed next. For this problem, we study approximation algorithms for non-identical machines. We develop a polynomial-time approximation scheme for unifo…
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 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…
This work introduces a natural variant of the online machine scheduling problem on unrelated machines, which we refer to as the favorite machine model. In this model, each job has a minimum processing time on a certain set of machines, called favorite machines, and some longer processing times on other machines. This type of costs (processing times) arise quite naturally in many practical problems. In the online version, jobs arrive one by one and must be allocated irrevocably upon each arrival without knowing the future jobs. We consider online algorithms for allocating jobs in order to minim…
In this paper, we consider the problem of choosing a minimum cost set of resources for executing a specified set of jobs. Each input job is an interval, determined by its start-time and end-time. Each resource is also an interval determined by its start-time and end-time; moreover, every resource has a capacity and a cost associated with it. We consider two versions of this problem. In the partial covering version, we are also given as input a number k, specifying the number of jobs that must be performed. The goal is to choose k jobs and find a minimum cost set of resources to perform the cho…
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.
In a classical scheduling problem, we are given a set of $n$ jobs of unit length along with precedence constraints and the goal is to find a schedule of these jobs on $m$ identical machines that minimizes the makespan. This problem is well-known to be NP-hard for an unbounded number of machines. Using standard 3-field notation, it is known as $P|\text{prec}, p_j=1|C_{\max}$. We present an algorithm for this problem that runs in $O(1.995^n)$ time. Before our work, even for $m=3$ machines the best known algorithms ran in $O^\ast(2^n)$ time. In contrast, our algorithm works when the number of mac…
The speed-robust scheduling problem is a two-stage problem where given $m$ machines, jobs must be grouped into at most $m$ bags while the processing speeds of the given $m$ machines are unknown. After the speeds are revealed, the grouped jobs must be assigned to the machines without being separated. To evaluate the performance of algorithms, we determine upper bounds on the worst-case ratio of the algorithm's makespan and the optimal makespan given full information. We refer to this ratio as the robustness factor. We give an algorithm with a robustness factor $2-1/m$ for the most general setti…
Weighted flow time is a fundamental and very well-studied objective function in scheduling. In this paper, we study the setting of a single machine with preemptions. The input consists of a set of jobs, characterized by their processing times, release times, and weights and we want to compute a (possibly preemptive) schedule for them. The objective is to minimize the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its release date and its completion time. It had been a long-standing open problem to find a polynomial time $O(1)$-approximation algorit…
Scheduling theory is an old and well-established area in combinatorial optimization, whereas the much younger area of parameterized complexity has only recently gained the attention of the community. Our aim is to bring these two areas closer together by studying the parameterized complexity of a class of single-machine two-agent scheduling problems. Our analysis focuses on the case where the number of jobs belonging to the second agent is considerably smaller than the number of jobs belonging to the first agent, and thus can be considered as a fixed parameter k. We study a variety of combinat…
Interval scheduling is a basic problem in the theory of algorithms and a classical task in combinatorial optimization. We develop a set of techniques for partitioning and grouping jobs based on their starting and ending times, that enable us to view an instance of interval scheduling on many jobs as a union of multiple interval scheduling instances, each containing only a few jobs. Instantiating these techniques in dynamic and local settings of computation leads to several new results. For $(1+\varepsilon)$-approximation of job scheduling of $n$ jobs on a single machine, we develop a fully dyn…
This paper establishes performance guarantees for online algorithms that schedule stochastic, nonpreemptive jobs on unrelated machines to minimize the expected total weighted completion time. Prior work on unrelated machine scheduling with stochastic jobs was restricted to the offline case, and required linear or convex programming relaxations for the assignment of jobs to machines. The algorithms introduced in this paper are purely combinatorial. The performance bounds are of the same order of magnitude as those of earlier work, and depend linearly on an upper bound on the squared coefficient…
Machine scheduling is a fundamental optimization problem in computer science. The task of scheduling a set of jobs on a given number of machines and minimizing the makespan is well studied and among other results, we know that EPTAS's for machine scheduling on identical machines exist. Das and Wiese initiated the research on a generalization of makespan minimization, that includes so called bag-constraints. In this variation of machine scheduling the given set of jobs is partitioned into subsets, so called bags. Given this partition a schedule is only considered feasible when on any machine th…
