Scheduling to Minimize Total Weighted Completion Time via Time-Indexed Linear Programming Relaxations

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. The recent breakthrough result of [Bansal-Srinivasan-Svensson, STOC 2016] gave a $(1.5-c)$-approximation for the problem, based on some lift-and-project SDP relaxation. Our main result is that a $(1.5 - c)$-approximation can also be achieved using a natural and considerably simpler time-indexed LP relaxation for the problem. We hope this relaxation can provide new insights into the problem

Scheduling on Hybrid Platforms: Improved Approximability Window

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 on the approximation ratio. These results improve upon the 6-approximation algorithm proposed by Kedad-Sidhoum et al. as well as the lower bound of 2 due to Svensson for identical machines. Our algorithm is inspired by the former one and distinguishes the allocation and the scheduling phases. However, we propose a different allocation procedure which, although is less efficient for the allocation sub-problem, leads to an improved approximation ratio for the whole scheduling problem. This approximation ratio actually decreases when the number of processing units of each type is close and matches the conditional lower bound when they are equal

Approximation Algorithms for Scheduling with Resource and Precedence Constraints

We study non-preemptive scheduling problems on identical parallel machines and uniformly related machines under both resource constraints and general precedence constraints between jobs. Our first result is an O(logn)-approximation algorithm for the objective of minimizing the makespan on parallel identical machines under resource and general precedence constraints. We then use this result as a subroutine to obtain an O(logn)-approximation algorithm for the more general objective of minimizing the total weighted completion time on parallel identical machines under both constraints. Finally, we present an O(logm logn)-approximation algorithm for scheduling under these constraints on uniformly related machines. We show that these results can all be generalized to include the case where each job has a release time. This is the first upper bound on the approximability of this class of scheduling problems where both resource and general precedence constraints must be satisfied simultaneously

On the Complexity of Conditional DAG Scheduling in Multiprocessor Systems

As parallel processing became ubiquitous in modern computing systems, parallel task models have been proposed to describe the structure of parallel applications. The workflow scheduling problem has been studied extensively over past years, focusing on multiprocessor systems and distributed environments (e.g. grids, clusters). In workflow scheduling, applications are modeled as directed acyclic graphs (DAGs). DAGs have also been introduced in the real-time scheduling community to model the execution of multi-threaded programs on a multi-core architecture. The DAG model assumes, in most cases, a fixed DAG structure capturing only straight-line code. Only recently, more general models have been proposed. In particular, the conditional DAG model allows the presence of control structures such as conditional (if-then-else) constructs. While first algorithmic results have been presented for the conditional DAG model, the complexity of schedulability analysis remains wide open. We perform a thorough analysis on the worst-case makespan (latest completion time) of a conditional DAG task under list scheduling (a.k.a. fixed-priority scheduling). We show several hardness results concerning the complexity of the optimization problem on multiple processors, even if the conditional DAG has a well-nested structure. For general conditional DAG tasks, the problem is intractable even on a single processor. Complementing these negative results, we show that certain practice-relevant DAG structures are very well tractable

Flow-time Optimization for Concurrent Open-Shop and Precedence Constrained Scheduling Models

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, broadcast scheduling, multi-dimensional scheduling, to name a few. Yet, our understanding of the flow-time objective is limited mostly to the scenarios where jobs have no dependencies. On the other hand, in almost all real world applications, think of MapReduce settings for example, jobs have dependencies that need to be respected while making scheduling decisions. In this paper, we take first steps towards understanding this complex problem. In particular, we consider two classical scheduling problems that capture dependencies across jobs: 1) concurrent open-shop scheduling (COSSP) and 2) precedence constrained scheduling. Our main motivation to study these problems specifically comes from their relevance to two scheduling problems that have gained importance in the context of data centers: co-flow scheduling and DAG scheduling. We design almost optimal approximation algorithms for COSSP and PCSP, and show hardness results

Communication-Aware Scheduling of Precedence-Constrained Tasks on Related Machines

Scheduling precedence-constrained tasks is a classical problem that has been studied for more than fifty years. However, little progress has been made in the setting where there are communication delays between tasks. Results for the case of identical machines were derived nearly thirty years ago, and yet no results for related machines have followed. In this work, we propose a new scheduler, Generalized Earliest Time First (GETF), and provide the first provable, worst-case approximation guarantees for the goals of minimizing both the makespan and total weighted completion time of tasks with precedence constraints on related machines with machine-dependent communication times