In this paper we study a single machine scheduling problem with the objective of minimizing the sum of completion times. Each of the given jobs is either short or long. However the processing times are initially hidden to the algorithm, but can be tested. This is done by executing a processing time oracle, which reveals the processing time of a given job. Each test occupies a time unit in the schedule, therefore the algorithm must decide for which jobs it will call the processing time oracle. The objective value of the resulting schedule is compared with the objective value of an optimal schedule, which is computed using full information. The resulting competitive ratio measures the price of hidden processing times, and the goal is to design an algorithm with minimal competitive ratio.
Two models are studied in this paper. In the non-adaptive model, the algorithm needs to decide beforehand which jobs to test, and which jobs to execute untested. However in the adaptive model, the algorithm can make these decisions adaptively depending on the outcomes of the job tests. In both models we provide optimal polynomial time algorithms following a two-phase strategy, which consist of a first phase where jobs are tested, and a second phase where jobs are executed obliviously. Experiments give strong evidence that optimal algorithms have this structure. Proving this property is left as an open problem.
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.
In this paper, we design the first streaming algorithms for the problem of multitasking scheduling on parallel machines with shared processing. In one pass, our streaming approximation schemes can provide an approximate value of the optimal makespan. If the jobs can be read in two passes, the algorithm can find the schedule with the approximate value. This work not only provides an algorithmic big data solution for the studied problem, but also gives an insight into the design of streaming algorithms for other problems in the area of scheduling.
We consider the problem of online scheduling on a single machine in order to minimize weighted flow time. The existing algorithms for this problem (STOC '01, SODA '03, FOCS '18) all require exact knowledge of the processing time of each job. This assumption is crucial, as even a slight perturbation of the processing time would lead to polynomial competitive ratio. However, this assumption very rarely holds in real-life scenarios. In this paper, we present the first algorithm for weighted flow time which do not require exact knowledge of the processing times of jobs. Specifically, we introduce …
Recently, the problem of multitasking scheduling has attracted a lot of attention in the service industries where workers frequently perform multiple tasks by switching from one task to another. Hall, Leung and Li (Discrete Applied Mathematics 2016) proposed a shared processing multitasking scheduling model which allows a team to continue to work on the primary tasks while processing the routinely scheduled activities as they occur. The processing sharing is achieved by allocating a fraction of the processing capacity to routine jobs and the remaining fraction, which we denote as sharing ratio…
Shortest Remaining Processing Time (SRPT) is a well known preemptive scheduling algorithm for uniprocessor and multiprocessor systems. SRPT finds applications in the emerging areas such as scheduling of client's requests that are submitted to a web server for accessing static web pages, managing the access requests to files in multiuser database systems and routing of packets across several links as per bandwidth availability in data communications. SRPT has been proved to be optimal for the settings, where the objective is to minimize the mean response time of a list of jobs. According to our…
We give a $(1+ε)$-approximate distance oracle with $O(1)$ query time for an undirected planar graph $G$ with $n$ vertices and non-negative edge lengths. For $ε>0$ and any two vertices $u$ and $v$ in $G$, our oracle gives a distance $\tilde{d}(u,v)$ with stretch $(1+ε)$ in $O(1)$ time. The oracle has size $O(n\log n ((\log n)/ε+f(ε)))$ and pre-processing time $O(n\log n((\log^3 n)/ε^2+f(ε)))$, where $f(ε)=2^{O(1/ε)}$. This is the first $(1+ε)$-approximate distance oracle with $O(1)$ query time independent of $ε$ and the size and pre-processing time nearly linear in $n$, and improves the query t…
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…
We study the generalized load-balancing (GLB) problem, where we are given $n$ jobs, each of which needs to be assigned to one of $m$ unrelated machines with processing times $\{p_{ij}\}$. Under a job assignment $σ$, the load of each machine $i$ is $ψ_i(\mathbf{p}_{i}[σ])$ where $ψ_i:\mathbb{R}^n\rightarrow\mathbb{R}_{\geq0}$ is a symmetric monotone norm and $\mathbf{p}_{i}[σ]$ is the $n$-dimensional vector $\{p_{ij}\cdot \mathbf{1}[σ(j)=i]\}_{j\in [n]}$. Our goal is to minimize the generalized makespan $ϕ(\mathsf{load}(σ))$, where $ϕ:\mathbb{R}^m\rightarrow\mathbb{R}_{\geq0}$ is another symmet…
We consider the classical minimum makespan scheduling problem, where the processing time of job $j$ on machine $i$ is $p_{ij}$, and the matrix $P=(p_{ij})_{m\times n}$ is of a low rank. It is proved in (Bhaskara et al., SODA 2013) that rank 7 scheduling is NP-hard to approximate to a factor of $3/2-ε$, and rank 4 scheduling is APX-hard (NP-hard to approximate within a factor of $1.03-ε$). We improve this result by showing that rank 4 scheduling is already NP-hard to approximate within a factor of $3/2-ε$, and meanwhile rank 3 scheduling is APX-hard.
We study the problem of preemptive scheduling n jobs with given release times on m identical parallel machines. The objective is to minimize the average flow time. We show that when all jobs have equal processing times then the problem can be solved in polynomial time using linear programming. Our algorithm can also be applied to the open-shop problem with release times and unit processing times. For the general case (when processing times are arbitrary), we show that the problem is unary NP-hard.
Most practical scheduling applications involve some uncertainty about the arriving times and lengths of the jobs. Stochastic online scheduling is a well-established model capturing this. Here the arrivals occur online, while the processing times are random. For this model, Gupta, Moseley, Uetz, and Xie recently devised an efficient policy for non-preemptive scheduling on unrelated machines with the objective to minimize the expected total weighted completion time. We improve upon this policy by adroitly combining greedy job assignment with $α_j$-point scheduling on each machine. In this way we…
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…
Modern platforms are using accelerators in conjunction with standard processing units in order to reduce the running time of specific operations, such as matrix operations, and improve their performance. Scheduling on such hybrid platforms is a challenging problem since the algorithms used for the case of homogeneous resources do not adapt well. In this paper we consider the problem of scheduling a set of tasks subject to precedence constraints on hybrid platforms, composed of two types of processing units. We propose a $(3+2\sqrt{2})$-approximation algorithm and a conditional lower bound of 3…
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…
In this work, we study the single machine scheduling problem with uncertain release times and processing times of jobs. We adopt a robust scheduling approach, in which the measure of robustness to be minimized for a given sequence of jobs is the worst-case objective function value from the set of all possible realizations of release and processing times. The objective function value is the total flow time of all jobs. We discuss some important properties of robust schedules for zero and non-zero release times, and illustrate the added complexity in robust scheduling given non-zero release time…
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 consider the classic problem of scheduling a set of n jobs non-preemptively on a single machine. Each job j has non-negative processing time, weight, and deadline, and a feasible schedule needs to be consistent with chain-like precedence constraints. The goal is to compute a feasible schedule that minimizes the sum of penalties of late jobs. Lenstra and Rinnoy Kan [Annals of Disc. Math., 1977] in their seminal work introduced this problem and showed that it is strongly NP-hard, even when all processing times and weights are 1. We study the approximability of the problem and our main result …
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…
Online Flexible Busy Time Scheduling on Heterogeneous Machines
Research Institute of the University of Bucharest（布加勒斯特大学研究所）；Northwestern University（西北大学）；Illinois Institute of Technology（伊利诺伊理工学院）；Department of EECS and the Simons Institute for the Theory of Computing, UC Berkeley（加州大学伯克利分校电气工程和计算机科学系及西蒙斯理论计算研究所）；Department of Computer Science, Northwestern University（西北大学计算机科学系）；Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学保罗森工程与应用科学学院）
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 online scheduling, jobs are available one by one and each job must be scheduled irrevocably before the availability of the next job. Semi-online scheduling is a relaxed variant of online scheduling, where an additional memory in terms of buffer or an Extra Piece of Information(EPI) is provided along with input data. The EPI may include one or more of the parameter(s) such as size of the largest job, total size of all jobs, arrival sequence of the jobs, optimum makespan value or range of job's processing time. A semi-online scheduling algorithm was first introduced in 1997 by Kellerer et al.…
We introduce a novel adversarial model for scheduling with explorable uncertainty. In this model, the processing time of a job can potentially be reduced (by an a priori unknown amount) by testing the job. Testing a job $j$ takes one unit of time and may reduce its processing time from the given upper limit $\bar{p}_j$ (which is the time taken to execute the job if it is not tested) to any value between $0$ and $\bar{p}_j$. This setting is motivated e.g. by applications where a code optimizer can be run on a job before executing it. We consider the objective of minimizing the sum of completion…
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…
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 this paper, we consider the problem of scheduling jobs on parallel identical machines, where the processing times of jobs are uncertain: only interval bounds of processing times are known. The optimality criterion of a schedule is the total completion time. In order to cope with the uncertainty, we consider the maximum regret objective and we seek a schedule that performs well under all possible instantiations of processing times. Although the deterministic version of the considered problem is solvable in polynomial time, the minmax regret version is known to be weakly NP-hard even for a si…
Serial-batch scheduling to minimise the total weighted late work
College of Economics and Management, Nanjing University of Aeronautics and Astronautics（南京航空航天大学经济与管理学院）；Institute of Computing Science, Poznan University of Technology（波兹南理工大学计算科学研究所）
Parallel machine scheduling has been extensively studied in the past decades, with applications ranging from production planning to job processing in large computing clusters. In this work we study some of these fundamental optimization problems, as well as their parameterized and reoptimization variants. We first present improved bounds for job scheduling on unrelated parallel machines, with the objective of minimizing the latest completion time (makespan) of the schedule. We consider the subclass of fully-feasible instances, in which the processing time of each job, on any machine, does not …
In the Scheduling Machines with Capacity Constraints problem, we are given k identical machines, each of which can process at most m_i jobs. M jobs are also given, where job j has a non-negative processing time length t_j >= 0. The task is to find a schedule such that the makespan is minimized and the capacity constraints are met. In this paper, we present a 3-approximation algorithm using an extension of Iterative Rounding Method introduced by Jain. To the best of the authors' knowledge, this is the first attempt to apply Iterative Rounding Method to scheduling problem with capacity constrain…
We study the restricted case of Scheduling on Unrelated Parallel Machines. In this problem, we are given a set of jobs $J$ with processing times $p_j$ and each job may be scheduled only on some subset of machines $S_j \subseteq M$. The goal is to find an assignment of jobs to machines to minimize the time by which all jobs can be processed. In a seminal paper, Lenstra, Shmoys, and Tardos designed an elegant $2$-approximation for the problem in 1987. The question of whether approximation algorithms with better guarantees exist for this classic scheduling problem has since remained a source of m…
Tree path minimum query problem is a fundamental problem while processing trees, and is used widely in minimum spanning tree verification and randomized minimum spanning tree algorithms. In this paper, we study the possibility of building an oracle in advance, which is able to answer the queries efficiently. We present an algorithm based on Boruvka trees. Our algorithm is the first to achieve a near-optimal bound on query time, while matching the currently optimal trade-off between construction time and the number of comparisons required at query. Particularly, in order to answer each query wi…
This paper addresses a single machine scheduling problem with non-preemptive jobs to minimize the total electricity cost. Two latest trends in the area of the energy-aware scheduling are considered, namely the variable energy pricing and the power-saving states of a machine. Scheduling of the jobs and the machine states are considered jointly to achieve the highest possible savings. Although this problem has been previously addressed in the literature, the reported results of the state-of-the-art method show that the optimal solutions can be found only for instances with up to 35 jobs and 209 …
It is well known that size-based scheduling policies, which take into account job size (i.e., the time it takes to run them), can perform very desirably in terms of both response time and fairness. Unfortunately, the requirement of knowing a priori the exact job size is a major obstacle which is frequently insurmountable in practice. Often, it is possible to get a coarse estimation of job size, but unfortunately analytical results with inexact job sizes are challenging to obtain, and simulation-based studies show that several size-based algorithm are severely impacted by job estimation errors.…
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.
Minimizing the sum of weighted completion times on $m$ identical parallel machines is one of the most important and classical scheduling problems. For the stochastic variant where processing times of jobs are random variables, Möhring, Schulz, and Uetz (1999) presented the first and still best known approximation result achieving, for arbitrarily many machines, performance ratio $1+\frac12(1+Δ)$, where $Δ$ is an upper bound on the squared coefficient of variation of the processing times. We prove performance ratio $1+\frac12(\sqrt{2}-1)(1+Δ)$ for the same underlying algorithm---the Weighted Sh…
We study the shared processor scheduling problem with a single shared processor where a unit time saving (weight) obtained by processing a job on the shared processor depends on the job. A polynomial-time optimization algorithm has been given for the problem with equal weights in the literature. This paper extends that result by showing an $O(n \log n)$ optimization algorithm for a class of instances in which non-decreasing order of jobs with respect to processing times provides a non-increasing order with respect to weights --- this instance generalizes the unweighted case of the problem. Thi…
Community networks (CNs) have seen an increase in the last fifteen years. Their members contact nodes which operate Internet proxies, web servers, user file storage and video streaming services, to name a few. Detecting communities of nodes with properties (such as co-location) and assessing node eligibility for service placement is thus a key-factor in optimizing the experience of users. We present a novel solution for the problem of service placement as a two-phase approach, based on: 1) community finding using a scalable graph label propagation technique and 2) a decentralized election proc…
In non-clairvoyant scheduling, the task is to find an online strategy for scheduling jobs with a priori unknown processing requirements with the objective to minimize the total (weighted) completion time. We revisit this well-studied problem in a recently popular learning-augmented setting that integrates (untrusted) predictions in online algorithm design. While previous works used predictions on processing requirements, we propose a new prediction model, which provides a relative order of jobs which could be seen as predicting algorithmic actions rather than parts of the unknown input. We sho…
This paper studies a single-machine scheduling problem with a non-renewable resource (NR-SSP) and total weighted completion time criterion. The non-renewable resource is consumed when the machine starts processing a job. We consider the case where each job's weight in the objective function is proportional to its resource consumption amount. The problem is known to be NP-hard in this case. We propose a 3-approximation list scheduling algorithm for this problem. Besides, we show that the approximation ratio 3 is tight for the algorithm.
In this study, we investigated several online and semi-online scheduling problems on two hierarchical machines with a common due date to maximize the total early work. For the pure online case, we designed an optimal online algorithm with a competitive ratio of $\sqrt 2$. For the case when the total processing time is known, we proposed an optimal semi-online algorithm with a competitive ratio of $\frac{4}{3}$. Additionally, for the cases when the largest processing time is known, we gave optimal algorithms with a competitive ratio of $\frac{6}{5}$ if the largest job is a lower hierarchy one, …
Online scheduling has been a well studied and challenging research problem over the last five decades since the pioneering work of Graham with immense practical significance in various applications such as interactive parallel processing, routing in communication networks, distributed data management, client-server communications, traffic management in transportation, industrial manufacturing and production. In this problem, a sequence of jobs is received one by one in order by the scheduler for scheduling over a number of machines. On arrival of a job, the scheduler assigns the job irrevocabl…
We investigate the recoverable robust single machine scheduling problem under interval uncertainty. In this setting, jobs have first-stage processing times p and second-stage processing times q and we aim to find a first-stage and second-stage schedule with a minimum combined sum of completion times, such that at least Delta jobs share the same position in both schedules. We provide positive complexity results for some important special cases of this problem, as well as derive a 2-approximation algorithm to the full problem. Computational experiments examine the performance of an exact mixed-i…
Columbia University in the City of New York（纽约市哥伦比亚大学）
Algorithms with predictions is a recent framework that has been used to overcome pessimistic worst-case bounds in incomplete information settings. In the context of scheduling, very recent work has leveraged machine-learned predictions to design algorithms that achieve improved approximation ratios in settings where the processing times of the jobs are initially unknown. In this paper, we study the speed-robust scheduling problem where the speeds of the machines, instead of the processing times of the jobs, are unknown and augment this problem with predictions. Our main result is an algorithm …
Motivated by mail delivery scheduling problems arising in Royal Mail, we study a generalization of the fundamental makespan scheduling P||Cmax problem which we call the bounded job start scheduling problem. Given a set of jobs, each specified by an integer processing time p_j, that have to be executed non-preemptively by a set of m parallel identical machines, the objective is to compute a minimum makespan schedule subject to an upper bound g<=m on the number of jobs that may simultaneously begin per unit of time. With perfect input knowledge, we show that Longest Processing Time First (LPT) a…
Parameterized Complexity of Scheduling Problems in Robotic Process Automation
Czech Technical University in Prague（布拉格捷克技术大学）；Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague（捷克技术大学捷克斯洛伐克信息学、机器人学与自动化研究所）；Sorbonne University（索邦大学）；CNRS（法国国家科学研究中心）
The paper considers single-machine scheduling problems with a non-renewable resource. In this setting, we are given a set jobs, each of which is characterized by a processing time, a weight, and the job also has some resource requirement. At fixed points in time, a certain amount of the resource is made available to be consumed by the jobs. The goal is to assign the jobs non-preemptively to time slots on the machine, so that at any time their resource requirement does not exceed the available amounts of resources. The objective that we consider here is the minimization of the sum of weighted c…
Distributed computing systems often need to consider the scheduling problem involving a collection of highly dependent data-processing tasks that must work in concert to achieve mission-critical objectives. This paper considers the unrelated machine scheduling problem for minimizing weighted sum completion time under arbitrary precedence constraints and on heterogeneous machines with different processing speeds. The problem is known to be strongly NP-hard even in the single machine setting. By making use of Queyranne's constraint set and constructing a novel Linear Programming relaxation for t…
In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardne…
We provide new (parameterized) computational hardness results for Interval Scheduling on Unrelated Machines. It is a classical scheduling problem motivated from just-in-time or lean manufacturing, where the goal is to complete jobs exactly at their deadline. We are given $n$ jobs and $m$ machines. Each job has a deadline, a weight, and a processing time that may be different on each machine. The goal is find a schedule that maximized the total weight of jobs completed exactly at their deadline. Note that this uniquely defines a processing time interval for each job on each machine. Interval Sc…
