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. Using the standard 3-field notation, it is known as $Pm|\text{prec}, p_j=1|C_{\max}$. Settling the complexity of $Pm|\text{prec}, p_j=1|C_{\max}$ even for $m=3$ machines is the last open problem from the book of Garey and Johnson [GJ79] for which both upper and lower bounds on the worst-case running times of exact algorithms solving them remain essentially unchanged since the publication of [GJ79]. We present an algorithm for this problem that runs in $(1+\frac{n}{m})^{\mathcal{O}(\sqrt{nm})}$ time. This algorithm is subexponential when $m = o(n)$. In the regime of $m=Θ(n)$ we show an algorithm that runs in$\mathcal{O}(1.997^n)$ time. Before our work, even for $m=3$ machines there were no algorithms known that run in $\mathcal{O}((2-\varepsilon)^n)$ time for some $\varepsilon > 0$.
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…
We consider the classic problem of scheduling jobs with precedence constraints on a set of identical machines to minimize the makespan objective function. Understanding the exact approximability of the problem when the number of machines is a constant is a well-known question in scheduling theory. Indeed, an outstanding open problem from the classic book of Garey and Johnson asks whether this problem is NP-hard even in the case of 3 machines and unit-length jobs. In a recent breakthrough, Levey and Rothvoss gave a $(1+ε)$-approximation algorithm, which runs in nearly quasi-polynomial time, for…
We consider a variant of the NP-hard problem of assigning jobs to machines to minimize the completion time of the last job. Usually, precedence constraints are given by a partial order on the set of jobs, and each job requires all its predecessors to be completed before it can start. In his seminal paper, Graham (1966) presented a simple 2-approximation algorithm, and, more than 40 years later, Svensson (2010) proved that 2 is essentially the best approximation ratio one can hope for in general. In this paper, we consider a different type of precedence relation that has not been discussed as e…
In a classical problem in scheduling, one has $n$ unit size jobs with a precedence order and the goal is to find a schedule of those jobs on $m$ identical machines as to minimize the makespan. It is one of the remaining four open problems from the book of Garey & Johnson whether or not this problem is $\mathbf{NP}$-hard for $m=3$. We prove that for any fixed $\varepsilon$ and $m$, an LP-hierarchy lift of the time-indexed LP with a slightly super poly-logarithmic number of $r = (\log(n))^{Θ(\log \log n)}$ rounds provides a $(1 + \varepsilon)$-approximation. For example Sherali-Adams suffices as…
We consider the problem of scheduling $n$ precedence-constrained jobs on $m$ uniformly-related machines in the presence of an arbitrary, fixed communication delay $ρ$. We consider a model that allows job duplication, i.e. processing of the same job on multiple machines, which, as we show, can reduce the length of a schedule (i.e., its makespan) by a logarithmic factor. Our main result is an $O(\log m \log ρ/ \log \log ρ)$-approximation algorithm for minimizing makespan, assuming the minimum makespan is at least $ρ$. Our algorithm is based on rounding a linear programming relaxation for the pro…
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.
A central problem in scheduling is to schedule $n$ unit size jobs with precedence constraints on $m$ identical machines so as to minimize the makespan. For $m=3$, it is not even known if the problem is NP-hard and this is one of the last open problems from the book of Garey and Johnson. We show that for fixed $m$ and $ε$, $(\log n)^{O(1)}$ rounds of Sherali-Adams hierarchy applied to a natural LP of the problem provides a $(1+ε)$-approximation algorithm running in quasi-polynomial time. This improves over the recent result of Levey and Rothvoss, who used $r=(\log n)^{O(\log \log n)}$ rounds of…
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 a natural variant of scheduling that we call \emph{partial scheduling}: In this variant an instance of a scheduling problem along with an integer $k$ is given and one seeks an optimal schedule where not all, but only $k$ jobs, have to be processed. Specifically, we aim to determine the fine-grained parameterized complexity of partial scheduling problems parameterized by $k$ for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints. That is, we inve…
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…
We study the classic problem of scheduling $n$ precedence constrained unit-size jobs on $m = O(1)$ machines so as to minimize the makespan. In a recent breakthrough, Levey and Rothvoss \cite{LR16} developed a $(1+ε)$-approximation for the problem with running time $\exp\Big(\exp\Big(O\big(\frac{m^2}{ε^2}\log^2\log n\big)\Big)\Big)$, via the Sherali-Adams lift of the basic linear programming relaxation for the problem by $\exp\Big(O\big(\frac{m^2}{ε^2}\log^2\log n\big)\Big)$ levels. Garg \cite{Garg18} recently improved the number of levels to $\log ^{O(m^2/ε^2)}n$, and thus the running time to …
We consider the problem of efficiently scheduling jobs with precedence constraints on a set of identical machines in the presence of a uniform communication delay. Such precedence-constrained jobs can be modeled as a directed acyclic graph, $G = (V, E)$. In this setting, if two precedence-constrained jobs $u$ and $v$, with $v$ dependent on $u$ ($u \prec v$), are scheduled on different machines, then $v$ must start at least $ρ$ time units after $u$ completes. The scheduling objective is to minimize makespan, i.e. the total time from when the first job starts to when the last job finishes. The f…
In the scheduling with non-uniform communication delay problem, the input is a set of jobs with precedence constraints. Associated with every precedence constraint between a pair of jobs is a communication delay, the time duration the scheduler has to wait between the two jobs if they are scheduled on different machines. The objective is to assign the jobs to machines to minimize the makespan of the schedule. Despite being a fundamental problem in theory and a consequential problem in practice, the approximability of scheduling problems with communication delays is not very well understood. On…
The scheduling literature has traditionally focused on a single type of resource (e.g., computing nodes). However, scientific applications in modern High-Performance Computing (HPC) systems process large amounts of data, hence have diverse requirements on different types of resources (e.g., cores, cache, memory, I/O). All of these resources could potentially be exploited by the runtime scheduler to improve the application performance. In this paper, we study multi-resource scheduling to minimize the makespan of computational workflows comprised of parallel jobs subject to precedence constraint…
This paper presents improved approximation algorithms for the problem of multiprocessor scheduling under uncertainty, or SUU, in which the execution of each job may fail probabilistically. This problem is motivated by the increasing use of distributed computing to handle large, computationally intensive tasks. In the SUU problem we are given n unit-length jobs and m machines, a directed acyclic graph G of precedence constraints among jobs, and unrelated failure probabilities q_{ij} for each job j when executed on machine i for a single timestep. Our goal is to find a schedule that minimizes th…
Consider a set of jobs connected to a directed acyclic task graph with a fixed source and sink. The edges of this graph model precedence constraints and the jobs have to be scheduled with respect to those. We introduce the Server Cloud Scheduling problem, in which the jobs have to be processed either on a single local machine or on one of many cloud machines. Both the source and the sink have to be scheduled on the local machine. For each job, processing times both on the server and in the cloud are given. Furthermore, for each edge in the task graph, a communication delay is included in the i…
We study the problem of scheduling precedence-constrained jobs on heterogenous machines in the presence of non-uniform job and machine communication delays. We are given as input $n$ unit size precedence-ordered jobs and $m$ related machines such that machine $i$ can execute up to $m_i$ jobs at a time. Each machine $i$ has an in-delay $ρ^{\mathrm{in}}_i$ and out-delay $ρ^{\mathrm{out}}_i$. Likewise, each job $v$ has an in-delay $ρ^{\mathrm{in}}_v$ and out-delay $ρ^{\mathrm{out}}_v$. In a schedule, job $v$ may be executed on machine $i$ at time $t$ if each predecessor $u$ of $v$ is completed on…
Inapproximability of Unique-Machine Precedence Scheduling for Unit-Length Jobs
Simons Institute for the Theory of Computing, and Departments of EECS & Mathematics, UC Berkeley（加州大学伯克利分校）；Department of EECS, UC Berkeley（加州大学伯克利分校）；Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）
Given unit execution time (UET) tasks whose precedence constraints form a directed acyclic graph (DAG), the arcs are associated with unit communication time (UCT) delays. The problem is to schedule the tasks on two processors in order to minimize the makespan. Several polynomial algorithms in the literature are proposed for special classes of digraphs, but the complexity of solving this problem in general case stills a challenging open question. We propose in this paper a linear time algorithm to compute an optimal schedule for the class of DAGs of depth two.
We consider the classic problem of scheduling jobs with precedence constraints on identical machines to minimize makespan, in the presence of communication delays. In this setting, denoted by $\mathsf{P} \mid \mathsf{prec}, c \mid C_{\mathsf{max}}$, if two dependent jobs are scheduled on different machines, then at least $c$ units of time must pass between their executions. Despite its relevance to many applications, this model remains one of the most poorly understood in scheduling theory. Even for a special case where an unlimited number of machines is available, the best known approximation…
A MILP model for an extended version of the Flexible Job Shop Scheduling problem is proposed. The extension allows the precedences between operations of a job to be given by an arbitrary directed acyclic graph rather than a linear order. The goal is the minimization of the makespan. Theoretical and practical advantages of the proposed model are discussed. Numerical experiments show the performance of a commercial exact solver when applied to the proposed model. The new model is also compared with a simple extension of the model described by Özgüven, Özbakir, and Yavuz (Mathematical models for …
University of Nottingham（诺丁汉大学）；University of Essex（埃塞克斯大学）
Multi-mode resource and precedence-constrained project scheduling is a well-known challenging real-world optimisation problem. An important variant of the problem requires scheduling of activities for multiple projects considering availability of local and global resources while respecting a range of constraints. A critical aspect of the benchmarks addressed in this paper is that the primary objective is to minimise the sum of the project completion times, with the usual makespan minimisation as a secondary objective. We observe that this leads to an expected different overall structure of goo…
