On the Complexity of Scheduling Unit-Time Jobs with OR-Precedence Constraints

AND/OR-networks are an important generalization of ordinary precedence constraints in various scheduling contexts. AND/OR-networks consist of traditional AND-precedence constraints, where a job can only be started after all its predecessors are completed, and OR-precedence constraints, where a job is ready as soon as any of its predecessors is completed. Hence, scheduling problems with AND/OR-constraints inherit the computational hardness of the corresponding problems with AND-precedence constraints. On the other hand, the complexity status of various scheduling problems with OR-constraints has remained open. In this paper, we present several complexity results for scheduling unit-time jobs subject to OR-precedence constraints. In particular, we give a polynomial-time algorithm for minimizing the makespan and the total completion time on identical parallel machines. This algorithm can also be applied if the number of available machines does not decrease over time. In the general case of profile scheduling, scheduling jobs with OR-precedence constraints to minimize the makespan or the total completion time is strongly NP-hard. Furthermore, it is not possible to approximate the makespan with a constant ratio, unless P=NP. In contrast to the makespan and the total completion time, minimizing the total weighted completion time is strongly NP-hard, even on a single machine

The Complexity of Mean Flow Time Scheduling Problems with Release Times

We study the problem of preemptive scheduling n jobs with given release times on m identical parallel machines. The objective is to minimize the average flow time. We show that when all jobs have equal processing times then the problem can be solved in polynomial time using linear programming. Our algorithm can also be applied to the open-shop problem with release times and unit processing times. For the general case (when processing times are arbitrary), we show that the problem is unary NP-hard.Comment: Subsumes and replaces cs.DS/0412094 and "Complexity of mean flow time scheduling problems with release dates" by P.B, S.

Restricted assignment scheduling with resource constraints

We consider parallel machine scheduling with job assignment restrictions, i.e., each job can only be processed on a certain subset of the machines. Moreover, each job requires a set of renewable resources. Any resource can be used by only one job at any time. The objective is to minimize the makespan. We present approximation algorithms with constant worst-case bound in the case that each job requires only a fixed number of resources. For some special cases optimal algorithms with polynomial running time are given. If any job requires at most one resource and the number of machines is fixed, we give a PTAS. On the other hand we prove that the problem is APX-hard, even when there are just three machines and the input is restricted to unit-time jobs. (C) 2018 Published by Elsevier B.V

How the structure of precedence constraints may change the complexity class of scheduling problems

This survey aims at demonstrating that the structure of precedence constraints plays a tremendous role on the complexity of scheduling problems. Indeed many problems can be NP-hard when considering general precedence constraints, while they become polynomially solvable for particular precedence constraints. We also show that there still are many very exciting challenges in this research area

Single-machine scheduling with stepwise tardiness costs and release times

We study a scheduling problem that belongs to the yard operations component of the railroad planning problems, namely the hump sequencing problem. The scheduling problem is characterized as a single-machine problem with stepwise tardiness cost objectives. This is a new scheduling criterion which is also relevant in the context of traditional machine scheduling problems. We produce complexity results that characterize some cases of the problem as pseudo-polynomially solvable. For the difficult-to-solve cases of the problem, we develop mathematical programming formulations, and propose heuristic algorithms. We test the formulations and heuristic algorithms on randomly generated single-machine scheduling problems and real-life datasets for the hump sequencing problem. Our experiments show promising results for both sets of problems

Minimizing the number of tardy jobs with precedence constraints and agreeable due dates

AbstractMinimizing the number of precedence constrained, unit-time tardy jobs is strongly NP-hard on a single machine. We study a special case of the problem where a job is tardy if it is finished more than a fixed K time units after its earliest possible completion time under the precedence constraints. We prove that the problem remains strongly NP-hard even with these special due dates. We also present polynomial time solutions for the weighted version of the problem if the precedence constraints are out-forests or interval orders. In the process, we also present a polynomial time solution for a special case of the minimum weight hitting set problem

Scheduling under Linear Constraints

We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job completion time among all feasible choices. This novel problem is motivated by various real-world application scenarios. We discuss the computational complexity and algorithms for various settings of this problem. In particular, we show that if there is only one machine with an arbitrary number of linear constraints, or there is an arbitrary number of machines with no more than two linear constraints, or both the number of machines and the number of linear constraints are fixed constants, then the problem is polynomial-time solvable via solving a series of linear programming problems. If both the number of machines and the number of constraints are inputs of the problem instance, then the problem is NP-Hard. We further propose several approximation algorithms for the latter case.Comment: 21 page

Complexity of scheduling multiprocessor tasks with prespecified processor allocations

We investigate the computational complexity of scheduling multiprocessor tasks with prespecified processor allocations. We consider two criteria: minimizing schedule length and minimizing the sum of the task completion times. In addition, we investigate the complexity of problems when precedence constraints or release dates are involved