Split Scheduling with Uniform Setup Times

We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine-, and sequence-independent. Problems of this kind were encountered when modelling practical problems in planning disaster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue why the same problem with three machines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with objective minimising weighted total completion time we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine-, and sequence-independent setup times

Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc

Split scheduling with uniform setup times

We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup, a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine- and sequence-independent. Problems of this kind were encountered when modelling practical problems in planning dis- aster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue, why the same problem with threemachines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with the objective to minimise total weighted completion time, we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine- and sequence-independent setup times

Malleable task-graph scheduling with a practical speed-up model

Scientific workloads are often described by Directed Acyclic task Graphs.Indeed, DAGs represent both a model frequently studied in theoretical literature and the structure employed by dynamic runtime schedulers to handle HPC applications. A natural problem is then to compute a makespan-minimizing schedule of a given graph. In this paper, we are motivated by task graphs arising from multifrontal factorizations of sparsematrices and therefore work under the following practical model. We focus on malleable tasks (i.e., a single task can be allotted a time-varying number of processors) and specifically on a simple yet realistic speedup model: each task can be perfectly parallelized, but only up to a limited number of processors. We first prove that the associated decision problem of minimizing the makespan is NP-Complete. Then, we study a widely used algorithm, PropScheduling, under this practical model and propose a new strategy GreedyFilling. Even though both strategies are 2-approximations, experiments on real and synthetic data sets show that GreedyFilling achieves significantly lower makespans

A statistical comparison of metaheuristics for unrelated parallel machine scheduling problems with setup times

Manufacturing scheduling aims to optimize one or more performance measures by allocating a set of resources to a set of jobs or tasks over a given period of time. It is an area that considers a very important decision-making process for manufacturing and production systems. In this paper, the unrelated parallel machine scheduling problem with machine-dependent and job-sequence-dependent setup times is addressed. This problem involves the scheduling of tasks on unrelated machines with setup times in order to minimize the makespan. The genetic algorithm is used to solve small and large instances of this problem when processing and setup times are balanced (Balanced problems), when processing times are dominant (Dominant P problems), and when setup times are dominant (Dominant S problems). For small instances, most of the values achieved the optimal makespan value, and, when compared to the metaheuristic ant colony optimization (ACOII) algorithm referred to in the literature, it was found that there were no significant differences between the two methods. However, in terms of large instances, there were significant differences between the optimal makespan obtained by the two methods, revealing overall better performance by the genetic algorithm for Dominant S and Dominant P problems.FCT—Fundação para a Ciência e Tecnologia through the R&D Units Project Scope UIDB/00319/2020 and EXPL/EME-SIS/1224/2021 and PhD grant UI/BD/150936/2021

Provably Efficient Adaptive Scheduling for Parallel Jobs

Scheduling competing jobs on multiprocessors has always been an important issue for parallel and distributed systems. The challenge is to ensure global, system-wide efficiency while offering a level of fairness to user jobs. Various degrees of successes have been achieved over the years. However, few existing schemes address both efficiency and fairness over a wide range of work loads. Moreover, in order to obtain analytical results, most of them require prior information about jobs, which may be difficult to obtain in real applications. This paper presents two novel adaptive scheduling algorithms -- GRAD for centralized scheduling, and WRAD for distributed scheduling. Both GRAD and WRAD ensure fair allocation under all levels of workload, and they offer provable efficiency without requiring prior information of job's parallelism. Moreover, they provide effective control over the scheduling overhead and ensure efficient utilization of processors. To the best of our knowledge, they are the first non-clairvoyant scheduling algorithms that offer such guarantees. We also believe that our new approach of resource request-allotment protocol deserves further exploration. Specifically, both GRAD and WRAD are O(1)-competitive with respect to mean response time for batched jobs, and O(1)-competitive with respect to makespan for non-batched jobs with arbitrary release times. The simulation results show that, for non-batched jobs, the makespan produced by GRAD is no more than 1.39 times of the optimal on average and it never exceeds 4.5 times. For batched jobs, the mean response time produced by GRAD is no more than 2.37 times of the optimal on average, and it never exceeds 5.5 times.Singapore-MIT Alliance (SMA

Group-based optimization for parallel job scheduling in clusters via heuristic search

Job scheduling for parallel processing typically makes scheduling decisions on a per job basis due to the dynamic arrival of jobs. Such decision making provides limited options to find globally best schedules. Most research uses off-line optimization which is not realistic. We propose an optimization on the basis of limited-size dynamic job grouping per priority class. We apply heuristic domain-knowledge-based hi-level search and branch-and-bound methods to heavy workload traces to capture good schedules. Special plan-based conservative backfilling and shifting policies are used to augment the search. Our objective is to minimize average relative response times for long and medium job classes, while keeping utilization high. The scheduling algorithm is extended from the SCOJO-PECT coarse-grain pre-emptive time-sharing scheduler. The proposed scheduler was evaluated using real traces and Lublin-Feitelson synthetic workload model. The comparisons were made with the conservative SCOJO-PECT scheduler. The results are promising--the average relative response times were improved by 18-32 while still able to contain the loss of utilization within 2

Improved Scheduling with a Shared Resource

We consider the following shared-resource scheduling problem: Given a set of jobs $J$, for each $j\in J$ we must schedule a job-specific processing volume of $v_j>0$. A total resource of $1$ is available at any time. Jobs have a resource requirement $r_j\in[0,1]$, and the resources assigned to them may vary over time. However, assigning them less will cause a proportional slowdown. We consider two settings. In the first, we seek to minimize the makespan in an online setting: The resource assignment of a job must be fixed before the next job arrives. Here we give an optimal $e/(e-1)$-competitive algorithm with runtime $\mathcal{O}(n\cdot \log n)$. In the second, we aim to minimize the total completion time. We use a continuous linear programming (CLP) formulation for the fractional total completion time and combine it with a previously known dominance property from malleable job scheduling to obtain a lower bound on the total completion time. We extract structural properties by considering a geometrical representation of a CLP's primal-dual pair. We combine the CLP schedule with a greedy schedule to obtain a $(3/2+\varepsilon)$-approximation for this setting. This improves upon the so far best-known approximation factor of $2$.Comment: Submitted to COCOA 2023, Full Versio