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

New complexity results for parallel identical machine scheduling problems with preemption, release dates and regular criteria

In this paper, we are interested in parallel identical machine scheduling problems with preemption and release dates in case of a regular criterion to be minimized. We show that solutions having a permutation flow shop structure are dominant if there exists an optimal solution with completion times scheduled in the same order as the release dates, or if there is no release date. We also prove that, for a subclass of these problems, the completion times of all jobs can be ordered in an optimal solution. Using these two results, we provide new results on polynomially solvable problems and hence refine the boundary between P and NP for these problems

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

Design and Analysis of an Estimation of Distribution Approximation Algorithm for Single Machine Scheduling in Uncertain Environments

In the current work we introduce a novel estimation of distribution algorithm to tackle a hard combinatorial optimization problem, namely the single-machine scheduling problem, with uncertain delivery times. The majority of the existing research coping with optimization problems in uncertain environment aims at finding a single sufficiently robust solution so that random noise and unpredictable circumstances would have the least possible detrimental effect on the quality of the solution. The measures of robustness are usually based on various kinds of empirically designed averaging techniques. In contrast to the previous work, our algorithm aims at finding a collection of robust schedules that allow for a more informative decision making. The notion of robustness is measured quantitatively in terms of the classical mathematical notion of a norm on a vector space. We provide a theoretical insight into the relationship between the properties of the probability distribution over the uncertain delivery times and the robustness quality of the schedules produced by the algorithm after a polynomial runtime in terms of approximation ratios

Preemptive scheduling on uniform parallel machines with controllable job processing times

In this paper, we provide a unified approach to solving preemptive scheduling problems with uniform parallel machines and controllable processing times. We demonstrate that a single criterion problem of minimizing total compression cost subject to the constraint that all due dates should be met can be formulated in terms of maximizing a linear function over a generalized polymatroid. This justifies applicability of the greedy approach and allows us to develop fast algorithms for solving the problem with arbitrary release and due dates as well as its special case with zero release dates and a common due date. For the bicriteria counterpart of the latter problem we develop an efficient algorithm that constructs the trade-off curve for minimizing the compression cost and the makespan

The robust single machine scheduling problem with uncertain release and processing times

In this work, we study the single machine scheduling problem with uncertain release times and processing times of jobs. We adopt a robust scheduling approach, in which the measure of robustness to be minimized for a given sequence of jobs is the worst-case objective function value from the set of all possible realizations of release and processing times. The objective function value is the total flow time of all jobs. We discuss some important properties of robust schedules for zero and non-zero release times, and illustrate the added complexity in robust scheduling given non-zero release times. We propose heuristics based on variable neighborhood search and iterated local search to solve the problem and generate robust schedules. The algorithms are tested and their solution performance is compared with optimal solutions or lower bounds through numerical experiments based on synthetic data

How Unsplittable-Flow-Covering helps Scheduling with Job-Dependent Cost Functions

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4-approximation algorithm. This is true even for a special case that is equivalent to the covering version of the well-studied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasi-polynomial time $(1+\epsilon)$-approximation algorithm that yields an $(e+\epsilon)$-approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with \emph{optimal }cost at $1+\epsilon$ speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most $\log n$ many classes. This algorithm allows the jobs even to have up to $\log n$ many distinct release dates.Comment: 2 pages, 1 figur