Approximation Schemes for Machine Scheduling

In the classical problem of makespan minimization on identical parallel machines, or machine scheduling for short, a set of jobs has to be assigned to a set of machines. The jobs have a processing time and the goal is to minimize the latest finishing time of the jobs. Machine scheduling is well known to be NP-hard and thus there is no polynomial time algorithm for this problem that is guaranteed to find an optimal solution unless P=NP. There is, however, a polynomial time approximation scheme (PTAS) for machine scheduling, that is, a family of approximation algorithms with ratios arbitrarily close to one. Whether a problem admits an approximation scheme or not is a fundamental question in approximation theory. In the present work, we consider this question for several variants of machine scheduling. We study the problem where the machines are partitioned into a constant number of types and the processing time of the jobs is also dependent on the machine type. We present so called efficient PTAS (EPTAS) results for this problem and variants thereof. We show that certain cases of machine scheduling with assignment restrictions do not admit a PTAS unless P=NP. Moreover, we introduce a graph framework based on the restrictions of the jobs and use it in the design of approximation schemes for other variants. We introduce an enhanced integer programming formulation for assignment problems, show that it can be efficiently solved, and use it in the EPTAS design for variants of machine scheduling with setup times. For one of the problems, we show that there is also a PTAS in the case with uniform machines, where machines have speeds influencing the processing times of the jobs. We consider cases in which each job requires a certain amount of a shared renewable resource and the processing time is depended on the amount of resource it receives or not. We present so called asymptotic fully polynomial time approximation schemes (AFPTAS) for the problems

Polynomial-time approximation schemes for scheduling problems with time lags

We identify two classes of machine scheduling problems with time lags that possess Polynomial-Time Approximation Schemes (PTASs). These classes together, one for minimizing makespan and one for minimizing total completion time, include many well-studied time lag scheduling problems. The running times of these approximation schemes are polynomial in the number of jobs, but exponential in the number of machines and the ratio between the largest time lag and the smallest positive operation time. These classes constitute the first PTAS results for scheduling problems with time lags

Sublinear Approximation Schemes for Scheduling Precedence Graphs of Bounded Depth

We study the classical scheduling problem on parallel machines %with precedence constraints where the precedence graph has the bounded depth h. Our goal is to minimize the maximum completion time. We focus on developing approximation algorithms that use only sublinear space or sublinear time. We develop the first one-pass streaming approximation schemes using sublinear space when all jobs\u27 processing times differ no more than a constant factor c and the number of machines m is at most 2nϵ3hc. This is so far the best approximation we can have in terms of m, since no polynomial time approximation better than 43 exists when m=n3 unless P=NP. %the problem cannot be approximated within a factor of 43 when m=n3 even if all jobs have equal processing time. The algorithms are then extended to the more general problem where the largest αn jobs have no more than c factor difference. % for some constant

On the Throughput of Large-but-Finite MIMO Networks using Schedulers

This paper studies the sum throughput of the {multi-user} multiple-input-single-output (MISO) networks in the cases with large but finite number of transmit antennas and users. Considering continuous and bursty communication scenarios with different users' data request probabilities, we derive quasi-closed-form expressions for the maximum achievable throughput of the networks using optimal schedulers. The results are obtained in various cases with different levels of interference cancellation. Also, we develop an efficient scheduling scheme using genetic algorithms (GAs), and evaluate the effect of different parameters, such as channel/precoding models, number of antennas/users, scheduling costs and power amplifiers' efficiency, on the system performance. Finally, we use the recent results on the achievable rates of finite block-length codes to analyze the system performance in the cases with short packets. As demonstrated, the proposed GA-based scheduler reaches (almost) the same throughput as in the exhaustive search-based optimal scheduler, with substantially less implementation complexity. Moreover, the power amplifiers' inefficiency and the scheduling delay affect the performance of the scheduling-based systems significantly