Yumei Huo

Bi-criteria scheduling problems: Number of tardy jobs and maximum weighted tardiness

Abstract Consider a single machine and a set of n jobs that are available for processing at time 0. Job j has a processing time p j , a due date d j and a weight w j . We consider bi-criteria scheduling problems involving the maximum weighted tardiness and the number of tardy jobs. We give NP-hardness proofs for the scheduling problems when either one of the two criteria is the primary criterion and the other one is the secondary criterion. These results answer two open questions posed by Lee and Vairaktarakis in 1993. We consider complexity relationships between the various problems, give polynomial-time algorithms for some special cases, and propose fast heuristics for the general case. The effectiveness of the heuristics is measured by empirical study. Our results show that one heuristic performs extremely well compared to optimal solutions.

Parallel machine scheduling with nested processing set restrictions

We consider the problem of scheduling a set of n independent jobs on m parallel machines, where each job can only be scheduled on a subset of machines called its processing set. The machines are linearly ordered, and the processing set of job j is given by two machine indexes aj and bj; i.e., job j can only be scheduled on machines aj,aj+1,...,bj. Two distinct processing sets are either nested or disjoint. Preemption is not allowed. Our goal is to minimize the makespan. It is known that the problem is strongly NP-hard and that there is a list-type algorithm with a worst-case bound of 2-1/m. In this paper we give an improved algorithm with a worst-case bound of 7/4. For two and three machines, the algorithm gives a better worst-case bound of 5/4 and 3/2, respectively.

Complexity of two dual criteria scheduling problems

In this article we answer the complexity question of two dual criteria scheduling problems which had been open for a long time. We show that both problems are binary NP-hard.

Fast approximation algorithms for job scheduling with processing set restrictions

We consider the problem of scheduling n independent jobs on m parallel machines, where the machines differ in their functionality but not in their processing speeds. Each job has a restricted set of machines to which it can be assigned, called its processing set. Preemption is not allowed. Our goal is to minimize the makespan of the schedule. We study two variants of this problem: (1) the case of tree-hierarchical processing set and (2) the case of nested processing set. We first give a fast algorithm for the case of tree-hierarchical processing set with a worst-case bound of 4/3, which is better than the best known algorithm whose worst-case bound is 2. We then give a more complicated algorithm for the case of nested processing set with a worst-case bound of 5/3, which is better than the best known algorithm whose worst-case bound is 7/4. In both cases, we will give examples achieving the worst-case bounds.

Exponential inapproximability and FPTAS for scheduling with availability constraints

We investigate the problems of scheduling n weighted jobs to one or more identical machines with the constraint that the machines may be unavailable in some specified time intervals. The objective is to find a schedule that minimizes the total weighted completion time. We consider both non-resumable and resumable schedules. Our first contributions concern approximability. For both resumable problem and non-resumable problem, we show that they cannot be approximated within an exponential factor by any polynomial time algorithm for multiple machines where each of them has an unavailable interval, even if the weight of each job equals to its processing time. Additionally, the non-resumable problem is also exponentially inapproximable for a single machine with two or more unavailable intervals. Then we develop the first FPTASs for the problems with a single unavailable interval among all machines. The running time is O(cnlog^dn(1@elogw)^d) for the non-resumable problem, and O(cnlog^dn(1@elogw)^d^+^1) for the resumable problem, where w is the product of the total weight and the total processing time of all jobs, c is the number of machines that are always available and d=6c+12. Thus our results give a clear boundary delineating the inapproximable cases and approximable cases. When there is a single machine and w=O(n^l^o^g^n^^^O^^^(^^^1^^^)), our algorithms greatly improve the current results. Note that instead of conventional ways of sequentially processing the jobs, our fast schemes process jobs in a divide-and-conquer fashion, which greatly reduces the running time. This may give some insight for some other related problems.

Online Scheduling of Precedence Constrained Tasks

A fundamental problem in scheduling theory is that of scheduling a set of n tasks, with precedence constraints, on

Total completion time minimization on multiple machines subject to machine availability and makespan constraints

m \ge 1

Minimizing total completion time in two-machine flow shops with exact delays

identical and parallel processors so as to minimize the makespan (schedule length). In the past, research has focused on the setting whereby all tasks are available for processing at the beginning (i.e., at time t = 0). In this article we consider the situation where tasks, along with their precedence constraints, are released at different times, and the scheduler has to make scheduling decisions without knowledge of future releases. In other words, the scheduler has to schedule tasks in an online fashion. We consider both preemptive and nonpreemptive schedules. We show that optimal online algorithms exist for some cases, while for others it is impossible to have one. Our results give a sharp boundary delineating the possible and the impossible cases.

Minimizing total weighted completion time with an unexpected machine unavailable interval

This paper studies preemptive bi-criteria scheduling on m parallel machines with machine unavailable intervals. The goal is to minimize the total completion time subject to the constraint that the makespan is at most a constant T. We study the unavailability model such that the number of available machines cannot go down by 2 within any period of pmax where pmax is the maximum processing time among all jobs. We show that there is an optimal polynomial time algorithm.

Online scheduling of equal-processing-time task systems

We study the problem of minimizing total completion time in the two-machine flow shop with exact delay model. This problem is a generalization of the no-wait flow shop problem which is known to be strongly NP-hard. Our problem has many applications but little results are given in the literature so far. We focus on permutation schedules. We first prove that some simple algorithms can be used to find the optimal schedules for some special cases. Then for the general case, we design some heuristics as well as metaheuristics whose performance are shown to be effective by computational experiments.