Joseph Y.-T. Leung

On the complexity of fixed-priority scheduling of periodic, real-time tasks

Abstract We consider the complexity of determining whether a set of periodic, real-time tasks can be scheduled on m ⩾ 1 identical processors with respect to fixed-priority scheduling. It is shown that the problem is NP-hard in all but one special case. The complexity of optimal fixed-priority scheduling algorithm is also discussed.

Minimizing Total Tardiness on One Machine is NP-Hard

The problem of minimizing the total tardiness for a set of independent jobs on one machine is considered. Lawler has given a pseudo-polynomial-time algorithm to solve this problem. In spite of extensive research efforts for more than a decade, the question of whether it can be solved in polynomial time or it is NP-hard in the ordinary sense remained open. In this paper the problem is shown to be NP-hard in the ordinary sense.

Complexity of scheduling parallel task systems

One of the assumptions made in classical scheduling theory is that a task is always executed by one processor at a time. With the advances in parallel algorithms, this assumption may not be valid for future task systems. In this paper, a new model of task systems is studied, the so-called Parallel Task System, in which a task can be executed by one or more processors at the same time. The complexity of scheduling Parallel Task Systems to minimize the schedule length is examined. For nonpreemptive scheduling, it is shown that the problem is strongly NP-hard even for two processors when the precedence constraints consist of a set of chains. For independent tasks, the problem is strongly NP-hard for five processors, but solvable in pseudo-polynomial time for two and three processors. For preemptive scheduling, it is shown that the problem is strongly NP-hard for arbitrary number of processors for a set of independent tasks. Furthermore, it is shown that it is NP-hard, but solvable in pseudo-polynomial time, ...

Efficient Algorithms for Interval Graphs and Circular-arc Graphs

We show that for an interval graph given in the form of a family of intervals, a maximum independent set, a minimum covering by disjoint completely connected sets or cliques, and a maximum clique can all be found in O(n log n) time [O(n) time if the endpoints of the intervals are sorted]. For the more general circular-arc graphs, a maximum independent set and a minimum covering by disjoint completely connected sets or cliques can be found in O(n2) time, provided again that a corresponding family of arcs is given.

Competitive Two-Agent Scheduling and Its Applications

We consider a scheduling environment with m (m ≥ 1) identical machines in parallel and two agents. Agent A is responsible for n1 jobs and has a given objective function with regard to these jobs; agent B is responsible for n2 jobs and has an objective function that may be either the same or different from the one of agent A. The problem is to find a schedule for the n1 + n2 jobs that minimizes the objective of agent A (with regard to his n1 jobs) while keeping the objective of agent B (with regard to his n2 jobs) below or at a fixed level Q. The special case with a single machine has recently been considered in the literature, and a variety of results have been obtained for two-agent models with objectives such as fmax, Σ wjCj, and Σ Uj. In this paper, we generalize these results and solve one of the problems that had remained open. Furthermore, we enlarge the framework for the two-agent scheduling problem by including the total tardiness objective, allowing for preemptions, and considering jobs with different release dates; we consider also identical machines in parallel. We furthermore establish the relationships between two-agent scheduling problems and other areas within the scheduling field, namely rescheduling and scheduling subject to availability constraints.

Packing squares into a square

Abstract The problem of determining whether a set of rectangles can be orthogonally packed into a larger rectangle has been studied as a geometric packing problem and as a floor planning problem. Recently, there is some renewed interest in this problem, as it is related to a job scheduling problem in a partitionable mesh connected system. In this paper we show that the problem of deciding whether a set of squares can be packed into a larger square is strongly NP-complete, answering an open question posed by Li and Cheng.

On a dual version of the one-dimensional bin packing problem

Abstract The NP-hard problem of packing items from a given set into bins so as to maximize the number of bins used, subject to the constraint that each bin be filled to at least a given threshold, is considered. Approximation algorithms are presented that provide guarantees of 1 2 , 2 3 , and 3 4 the optimal number, at running time costs of O(n), O(nlogn), and O(nlog2n), respectively, and the average case behavior of these algorithms is explored via empirical tests on randomly generated sets of items.

On-line scheduling of real-time tasks

An optimal on-line scheduler is given for a set of real-time tasks with one common deadline on m processors. It is shown that no optimal scheduler can exist for tasks with two distinct deadlines. Finally, an optimal on-line scheduler is given for situations where processors can go down unexpectedly. >

A new algorithm for scheduling periodic, real-time tasks

We consider the problem of preemptively scheduling a set of periodic, real-time tasks on a multiprocessor computer system. We give a new scheduling algorithm, the so-called Slack-Time Algorithm, and show that it is more effective than the known Deadline Algorithm. We also give an (exponential-time) algorithm to decide if a task system is schedulable by the Slack-Time or the Deadline Algorithm. The same algorithm can also be used to decide if a task system is schedulable by any given fixed-priority scheduling algorithm. This resolves an open question posed by Leung and Whitehead. Finally, it is shown that the problem of deciding if a task system is schedulable by the Slack-Time, the Deadline, or any given fixed-priority scheduling algorithm is co-NP-hard for each fixedm≥.

Complexity of scheduling tasks with time-dependent execution times

Abstract A new model of task system in which the execution time of a task depends on its starting time is considered. It is shown that the feasibility problem on a single processor is NP-complete for a set of tasks with identical release times and two distinct deadlines. An O(n log n)-time algorithm is given for a set of tasks with identical release times and deadlines. These two results give a sharp boundary delineating the complexity of the problem.