Tight Approximation Algorithms for Scheduling with Fixed Jobs and Non-Availability

We study two closely related problems in non-preemptive scheduling of sequential jobs on identical parallel machines. In these two settings there are either fixed jobs or non-availability intervals during which the machines are not available; in both cases, the objective is to minimize the makespan. Both formulations have different applications, e.g. in turnaround scheduling or overlay computing. For both problems we contribute approximation algorithms with an improved ratio of $3/2+\epsilon$, respectively, which we refine to approximation algorithms with ratio $3/2$. For scheduling with fixed jobs, a lower bound of $3/2$ on the approximation ratio has been obtained by Scharbrodt, Steger & Weisser: for scheduling with non-availability we provide the same lower bound. In total, our approximation ratio for both problems is tight via suitable inapproximability results. We use dual approximation, creation of a gap structure and job configurations, and a PTAS for the multiple subset sum problem. However, the main feature of our algorithms is a new technique for the assignment of large jobs via flexible rounding. Our new technique is based on an interesting cyclic shifting argument in combination with a network flow model for the assignment of jobs to large gaps

Tight Approximation Algorithms for Scheduling with Fixed Jobs and Non-Availability

We study two closely related problems in non-preemptive scheduling of jobs on identical parallel machines. In these two settings there are either fixed jobs or non-availability intervals during which the machines are not available; in both cases, the objective is to minimize the makespan. Both formulations have different applications, e.g. in turnaround scheduling or overlay computing. For both problems we contribute approximation algorithms with an improved ratio of 3/2. For scheduling with fixed jobs, a lower bound of 3/2 on the approximation ratio has been obtained by Scharbrodt, Steger & Weisser; for scheduling with non-availability we provide the same lower bound. We use dual approximation, creation of a gap structure and a PTAS for the multiple subset sum problem, combined with a post- processing step to assign large job

Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc

Stochastic scheduling on unrelated machines

Two important characteristics encountered in many real-world scheduling problems are heterogeneous machines/processors and a certain degree of uncertainty about the actual sizes of jobs. The first characteristic entails machine dependent processing times of jobs and is captured by the classical unrelated machine scheduling model.The second characteristic is adequately addressed by stochastic processing times of jobs as they are studied in classical stochastic scheduling models. While there is an extensive but separate literature for the two scheduling models, we study for the first time a combined model that takes both characteristics into account simultaneously. Here, the processing time of job $j$ on machine $i$ is governed by random variable $P_{ij}$, and its actual realization becomes known only upon job completion. With $w_j$ being the given weight of job $j$, we study the classical objective to minimize the expected total weighted completion time $E[\sum_j w_jC_j]$, where $C_j$ is the completion time of job $j$. By means of a novel time-indexed linear programming relaxation, we compute in polynomial time a scheduling policy with performance guarantee $(3+\Delta)/2+\epsilon$. Here, $\epsilon>0$ is arbitrarily small, and $\Delta$ is an upper bound on the squared coefficient of variation of the processing times. We show that the dependence of the performance guarantee on $\Delta$ is tight, as we obtain a $\Delta/2$ lower bound for the type of policies that we use. When jobs also have individual release dates $r_{ij}$, our bound is $(2+\Delta)+\epsilon$. Via $\Delta=0$, currently best known bounds for deterministic scheduling are contained as a special case

Competitive-Ratio Approximation Schemes for Minimizing the Makespan in the Online-List Model

We consider online scheduling on multiple machines for jobs arriving one-by-one with the objective of minimizing the makespan. For any number of identical parallel or uniformly related machines, we provide a competitive-ratio approximation scheme that computes an online algorithm whose competitive ratio is arbitrarily close to the best possible competitive ratio. We also determine this value up to any desired accuracy. This is the first application of competitive-ratio approximation schemes in the online-list model. The result proves the applicability of the concept in different online models. We expect that it fosters further research on other online problems

Throughput Maximization in Multiprocessor Speed-Scaling

We are given a set of $n$ jobs that have to be executed on a set of $m$ speed-scalable machines that can vary their speeds dynamically using the energy model introduced in [Yao et al., FOCS'95]. Every job $j$ is characterized by its release date $r_j$, its deadline $d_j$, its processing volume $p_{i,j}$ if $j$ is executed on machine $i$ and its weight $w_j$. We are also given a budget of energy $E$ and our objective is to maximize the weighted throughput, i.e. the total weight of jobs that are completed between their respective release dates and deadlines. We propose a polynomial-time approximation algorithm where the preemption of the jobs is allowed but not their migration. Our algorithm uses a primal-dual approach on a linearized version of a convex program with linear constraints. Furthermore, we present two optimal algorithms for the non-preemptive case where the number of machines is bounded by a fixed constant. More specifically, we consider: {\em (a)} the case of identical processing volumes, i.e. $p_{i,j}=p$ for every $i$ and $j$, for which we present a polynomial-time algorithm for the unweighted version, which becomes a pseudopolynomial-time algorithm for the weighted throughput version, and {\em (b)} the case of agreeable instances, i.e. for which $r_i \le r_j$ if and only if $d_i \le d_j$, for which we present a pseudopolynomial-time algorithm. Both algorithms are based on a discretization of the problem and the use of dynamic programming