A branch, bound, and remember algorithm for the 1|r i |∑t i scheduling problem

This paper presents a modified Branch and Bound (B&B) algorithm called, the Branch, Bound, and Remember (BB&R) algorithm, which uses the Distributed Best First Search (DBFS) exploration strategy for solving the 1|r _i |∑t _i scheduling problem, a single machine scheduling problem where the objective is to find a schedule with the minimum total tardiness. Memory-based dominance strategies are incorporated into the BB&R algorithm. In addition, a modified memory-based dynamic programming algorithm is also introduced to efficiently compute lower bounds for the 1|r _i |∑t _i scheduling problem. Computational results are reported, which shows that the BB&R algorithm with the DBFS exploration strategy outperforms the best known algorithms reported in the literature.     

Online Randomized Multiprocessor Scheduling

The use of randomization in online multiprocessor scheduling is studied. The problem of scheduling independent jobs on m machines online originates with Graham [16]. While the deterministic case of this problem has been studied extensively, little work has been done on the randomized case. For m= 2 a randomized 4/3-competitive algorithm was found by Bartal et al. A randomized algorithm for m ≥ 3 is presented. It achieves competitive ratios of 1.55665, 1.65888, 1.73376, 1.78295, and 1.81681, for m = 3, 4, 5, 6,7 , respectively. These competitive ratios are less than the best deterministic lower bound for m=3,4,5 and less than the best known deterministic competitive ratio for m = 3,4,5,6,7 . Two models of multiprocessor scheduling with rejection are studied. The first is the model of Bartal et al. Two randomized algorithms for this model are presented. The first algorithm performs well for small m , achieving competitive ratios of 3/2 , , for m=2,3,4 , respectively. The second algorithm outperforms the first for m ≥ 5 . It beats the deterministic algorithm of Bartal et al. for all m = 5 ,. . ., 1000 . It is conjectured that this is true for all m . The second model differs in that preemption is allowed. For this model, randomized algorithms are presented which outperform the best deterministic algorithm for small m . Received August 11, 1997; revised February 25, 1998.     

Dynamic TCP Acknowledgment and Other Stories about e/(e - 1)

Abstract. We present the first optimal randomized online algorithms for the TCP acknowledgment problem [3] and the Bahncard problem [5]. These problems are well known to be generalizations of the classical online ski-rental problem, however, they appeared to be harder. In this paper we demonstrate that a number of online algorithms which have optimal competitive ratios of e/(e-1) , including these, are fundamentally no more complex than ski rental. Our results also suggest a clear paradigm for solving ski-rental-like problems.     

An Experimental Study of Algorithms for Weighted Completion Time Scheduling

We consider the total weighted completion time scheduling problem for parallel identical machines and precedence constraints, P| prec|\sum w _i C _i . This important and broad class of problems is known to be NP-hard, even for restricted special cases, and the best known approximation algorithms have worst-case performance that is far from optimal. However, little is known about the experimental behavior of algorithms for the general problem. This paper represents the first attempt to describe and evaluate comprehensively a range of weighted completion time scheduling algorithms. We first describe a family of combinatorial scheduling algorithms that optimally solve the single-machine problem, and show that they can be used to achieve good performance for the multiple-machine problem. These algorithms are efficient and find schedules that are on average within 1.5\percent of optimal over a large synthetic benchmark consisting of trees, chains, and instances with no precedence constraints. We then present several ways to create feasible schedules from nonintegral solutions to a new linear programming relaxation for the multiple-machine problem. The best of these linear programming-based approaches finds schedules that are within 0.2\percent of optimal over our benchmark. Finally, we describe how the scheduling phase in profile-based program compilation can be expressed as a weighted completion time scheduling problem and apply our algorithms to a set of instances extracted from the SPECint95 compiler benchmark. For these instances with arbitrary precedence constraints, the best linear programming-based approach finds optimal solutions in 78\percent of cases. Our results demonstrate that careful experimentation can help lead the way to high quality algorithms, even for difficult optimization problems. Received October 30, 1998; revised March 28, 2001.     

Semi-online Machine Covering on Two Uniform Machines with Known Total Size

This paper investigates semi-online scheduling problem with known total size on two uniform machines for maximizing the minimum machine completion time. Lower bounds and optimal algorithms for every s≥1 are given, where s is the speed ratio of two machines. Both the overall competitive ratio and the competitive ratio for are strictly smaller than those of the optimal algorithms of the corresponding semi-online scheduling problem with known optimal value. It indicates that two related semi-online problems are different.     

Scheduling jobs with equal processing times subject to machine eligibility constraints

We consider the problem of nonpreemptively scheduling a set of n jobs with equal processing times on m parallel machines so as to minimize the makespan. Each job has a prespecified set of machines on which it can be processed, called its eligible set. We consider the most general case of machine eligibility constraints as well as special cases of nested and inclusive eligible sets. Both online and offline models are considered. For offline problems we develop optimal algorithms that run in polynomial time, while for online problems we focus on the development of optimal algorithms of a new and more elaborate structure as well as approximation algorithms with good competitive ratios.     

Scheduling a Single Server in a Two-machine Flow Shop

We study the problem of scheduling a single server that processes n jobs in a two-machine flow shop environment. A machine dependent setup time is needed whenever the server switches from one machine to the other. The problem with a given job sequence is shown to be reducible to a single machine batching problem. This result enables several cases of the server scheduling problem to be solved in O(n log n) by known algorithms, namely, finding a schedule feasible with respect to a given set of deadlines, minimizing the maximum lateness and, if the job processing times are agreeable, minimizing the total completion time. Minimizing the total weighted completion time is shown to be NP-hard in the strong sense. Two pseudopolynomial dynamic programming algorithms are presented for minimizing the weighted number of late jobs. Minimizing the number of late jobs is proved to be NP-hard even if setup times are equal and there are two distinct due dates. This problem is solved in O(n ³) time when all job processing times on the first machine are equal, and it is solved in O(n ⁴) time when all processing times on the second machine are equal. Received November 20, 2001; revised October 18, 2002 Published online: January 16, 2003     

Minimizing Makespan and Preemption Costs on a System of Uniform Machines

It is well known that for preemptive scheduling on uniform machines there exist polynomial time exact algorithms, whereas for non-preemptive scheduling there are probably no such algorithms. However, it is not clear how many preemptions (in total, or per job) suffice in order to guarantee an optimal polynomial time algorithm. In this paper we investigate exactly this hardness gap, formalized as two variants of the classic preemptive scheduling problem. In generalized multiprocessor scheduling (GMS) we have a job-wise or total bound on the number of preemptions throughout a feasible schedule. We need to find a schedule that satisfies the preemption constraints, such that the maximum job completion time is minimized. In minimum preemptions scheduling (MPS) the only feasible schedules are preemptive schedules with the smallest possible makespan. The goal is to find a feasible schedule that minimizes the overall number of preemptions. Both problems are NP-hard, even for two machines and zero preemptions. For GMS, we develop polynomial time approximation schemes, distinguishing between the cases where the number of machines is fixed, or given as part of the input. Our scheme for a fixed number of machines has linear running time, and can be applied also for instances where jobs have release dates, and for instances with arbitrary preemption costs. For MPS, we derive matching lower and upper bounds on the number of preemptions required by any optimal schedule. Our results for MPS hold for any instance in which a job, J_j, can be processed simultaneously by ρ_j machines, for some ρ_j ≥ 1.     

Sensitivity analysis in the single-machine scheduling problem with max–min criterion

In this paper, the single‐machine scheduling problem 1∣prec∣f_max is considered. It is one of the most general scheduling problems for which an efficient, polynomial algorithm has been developed. It is always possible to calculate quickly one optimal solution (a sequence of jobs) in that problem. However, the set of all optimal solutions may contain a lot of other sequences, so it is important to give a full characterization of that set. This paper consists of two parts. In the first part, some sufficient and necessary conditions of optimality of a given solution to the problem 1∣prec∣f_max are proved. In the second part, an application of these conditions to the sensitivity analysis is presented.     

Scheduling unit length jobs on parallel machines with lookahead information

This paper studies two closely related online-list scheduling problems of a set of n jobs with unit processing times on a set of m multipurpose machines. It is assumed that there are k different job types, where each job type can be processed on a unique subset of machines. In the classical definition of online-list scheduling, the scheduler has all the information about the next job to be scheduled in the list while there is uncertainty about all the other jobs in the list not yet scheduled. We extend this classical definition to include lookahead abilities, i.e., at each decision point, in addition to the information about the next job in the list, the scheduler has all the information about the next h jobs beyond the current one in the list. We show that for the problem of minimizing the makespan there exists an optimal (1-competitive) algorithm for the online problem when there are two job types. That is, the online algorithm gives the same minimal makespan as the optimal offline algorithm for any instance of the problem. Furthermore, we show that for more than two job types no such online algorithm exists. We also develop several dynamic programming algorithms to solve a stochastic version of the problem, where the probability distribution of the job types is known and the objective is to minimize the expected makespan.