Unrelated Machines Research Articles

Proportionally Fair Makespan Approximation

We study fair mechanisms for the classic job scheduling problem on unrelated machines with the objective of minimizing the makespan. This problem is equivalent to minimizing the egalitarian social cost in the fair division of chores. The two prevalent fairness notions in the fair division literature are envy-freeness and proportionality. Prior work has established that no envy-free mechanism can provide better than an Ω(log m / log log m)-approximation to the optimal makespan, where m is the number of machines, even when payments to the machines are allowed. In strong contrast to this impossibility, our main result demonstrates that there exists a proportional mechanism (with payments) that achieves a 3/2-approximation to the optimal makespan, and this ratio is tight. To prove this result, we provide a full characterization of allocation functions that can be made proportional with payments. Furthermore, we show that for instances with normalized costs, there exists a proportional mechanism that achieves the optimal makespan. We conclude with important directions for future research concerning other fairness notions, including relaxations of envy-freeness. Notably, we show that the technique leading to the impossibility result for envy-freeness does not extend to its relaxations.

Approximation algorithms for maximum weighted throughput on unrelated machines

Approximation algorithms for maximum weighted throughput on unrelated machines

Truthful Allocation in Graphs and Hypergraphs

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of truthful mechanisms that have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k -degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L p -norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ∞, and extend the results to any p > 0.

Complexity of scheduling few types of jobs on related and unrelated machines

The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number k of job types, but possibly the number of jobs n is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines (Q|HM|Cmax) is NP-hard already with 6 job types, and that the related Cutting Stock problem is NP-hard already with 8 item types. For the more general unrelated machines model (R|HM|Cmax), we show that if the largest job size pmax or the number of jobs n is polynomially bounded in the instance size |I|, there are algorithms with complexity |I|poly(k). Our main result is that this is unlikely to be improved because Q||Cmax is W[1]-hard parameterized by k already when n, pmax, and the numbers describing the machine speeds are polynomial in |I|; the same holds for R||Cmax (without machine speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives ℓ2-norm minimization of the load vector and, partially, sum of weighted completion times ∑wjCj. Along the way, we answer affirmatively the question whether makespan minimization on identical machines (P||Cmax) is fixed-parameter tractable parameterized by k, extending our understanding of this fundamental problem. Together with our hardness results for Q||Cmax, this implies that the complexity of P|HM|Cmax is the only remaining open case.

Permutation Predictions for Non-Clairvoyant Scheduling

In non-clairvoyant scheduling, the task is to schedule jobs with a priori unknown processing requirements. We revisit this well-studied problem with the objective of minimizing the total (weighted) completion time in a recently popular learning-augmented setting that integrates possibly imperfect predictions into online algorithm design. While previous works used predictions on processing requirements, we propose a new prediction model that provides a relative order of jobs, which could be seen as predicting algorithmic actions rather than parts of the unknown input. We show that these succinct predictions have desired properties, admit a natural error measure, and enable algorithms with strong performance guarantees. Additionally, these predictions are learnable in both theory and practice. We generalize the algorithmic framework proposed in the seminal paper by Purohit, Kumar, and Svitkina (NeurIPS 2018) and present the first learning-augmented scheduling results for weighted jobs and unrelated machines. We demonstrate in empirical experiments the practicability and superior performance compared to the previously suggested single-machine algorithms.

Automated generation of dispatching rules for the green unrelated machines scheduling problem

The concept of green scheduling, which deals with the environmental impact of the scheduling process, is becoming increasingly important due to growing environmental concerns. Most green scheduling problem variants focus on modelling the energy consumption during the execution of the schedule. However, the dynamic unrelated machines environment is rarely considered, mainly because it is difficult to manually design simple heuristics, called dispatching rules (DRs), which are suitable for solving dynamic, non-standard scheduling problems. Using hyperheuristics, especially genetic programming (GP), alleviates the problem since it enables the automatic design of new DRs. In this study, we apply GP to automatically design DRs for solving the green scheduling problem in the unrelated machines environment under dynamic conditions. The total energy consumed during the system execution is optimised along with two standard scheduling criteria. The three most commonly investigated green scheduling problem variants from the literature are selected, and GP is adapted to generate appropriate DRs for each. The experiments show that GP-generated DRs efficiently solve the problem under dynamic conditions, providing a trade-off between optimising standard and energy-related criteria.

On Scheduling Mechanisms Beyond the Worst Case

The problem of scheduling unrelated machines has been studied since the inception of algorithmic mechanism design (Nisan and Ronen, Algorithmic mechanism design(extended abstract). In: Proceedings of the Thirty First Annual ACM Symposium on Theory of Computing (STOC), pp. 129–140, 1999. It is a resource allocation problem that entails assigning m tasks to n machines for execution. Machines are regarded as strategic agents who may lie about their execution costs so as to minimize their time cost. To address the situation when monetary payment is not an option to compensate the machines’ costs, Koutsoupias (Theory Comput Syst 54:375–387, 2014) devised two truthful mechanisms, K and P respectively, that achieves an approximation ratio of n+12 and n, for social cost minimization. In addition, no truthful mechanism can achieve an approximation ratio better than n+12. Hence, mechanism K is optimal. While the approximation ratio provides a strong worst-case guarantee, it also limits us to a comprehensive understanding of mechanism performance on various inputs. This paper investigates these two scheduling mechanisms beyond the worst case. We first show that mechanism K achieves a smaller social cost than mechanism P on every input. That is, mechanism K is pointwise better than mechanism P. Next, for each task, when machines’ execution costs are independent and identically drawn from a task-specific distribution, we show that the average-case approximation ratio of mechanism K converges to a constant determined by the task-specific distribution. This bound is tight for mechanism K. For a better understanding of this distribution-dependent constant, on the one hand, we estimate its value by plugging in a few common distributions; on the other, we show that this converging bound improves a known bound (Zhang in Algorithmica 83(6):1638–1652, 2021)) which only captures the single-task setting. Last, we find that the average-case approximation ratio of mechanism P converges to the same constant.

Minimizing the maximum lateness for scheduling with release times and job rejection

We study scheduling problems with release times and rejection costs with the objective function of minimizing the maximum lateness. Our main result is a PTAS for the single machine problem with an upper bound on the rejection costs. This result is extended to parallel, identical machines. The corresponding problem of minimizing the rejection costs with an upper bound on the lateness is also examined. We show how to compute a PTAS for determining an approximation of the Pareto frontier on both objective functions on parallel, identical machines. Moreover, we present an FPTAS with strongly polynomial time for the maximum lateness problem without release times on identical machines when the number of machines is constant. Finally, we extend this FPTAS to the case of unrelated machines.

Configuration balancing for stochastic requests

The configuration balancing problem with stochastic requests generalizes well-studied resource allocation problems such as load balancing and virtual circuit routing. There are given m resources and n requests; each request has multiple possible configurations, each of which increases the load of each resource by some amount. The goal is to select one configuration for each request to minimize the makespan: the load of the most-loaded resource. In the stochastic setting, the amount by which a configuration increases the resource load is uncertain until the configuration is chosen, but we are given a probability distribution. We develop both offline and online algorithms for configuration balancing with stochastic requests. When the requests are known offline, we give a non-adaptive policy for configuration balancing with stochastic requests that O(logmloglogm)-approximates the optimal adaptive policy, which matches a known lower bound for the special case of load balancing on identical machines. When requests arrive online in a list, we give a non-adaptive policy that is O(logm) competitive. Again, this result is asymptotically tight due to information-theoretic lower bounds for special cases (e.g., for load balancing on unrelated machines). Finally, we show how to leverage adaptivity in the special case of load balancing on related machines to obtain a constant-factor approximation offline and an O(loglogm)-approximation online. A crucial technical ingredient in all of our results is a new structural characterization of the optimal adaptive policy that allows us to limit the correlations between its decisions.

Due-Date assignment with acceptable lead-times on parallel machines

Due-Date assignment with acceptable lead-times on parallel machines

Assessing the Ability of Genetic Programming for Feature Selection in Constructing Dispatching Rules for Unrelated Machine Environments

The automated design of dispatching rules (DRs) with genetic programming (GP) has become an important research direction in recent years. One of the most important decisions in applying GP to generate DRs is determining the features of the scheduling problem to be used during the evolution process. Unfortunately, there are no clear rules or guidelines for the design or selection of such features, and often the features are simply defined without investigating their influence on the performance of the algorithm. However, the performance of GP can depend significantly on the features provided to it, and a poor or inadequate selection of features for a given problem can result in the algorithm performing poorly. In this study, we examine in detail the features that GP should use when developing DRs for unrelated machine scheduling problems. Different types of features are investigated, and the best combination of these features is determined using two selection methods. The obtained results show that the design and selection of appropriate features are crucial for GP, as they improve the results by about 7% when only the simplest terminal nodes are used without selection. In addition, the results show that it is not possible to outperform more sophisticated manually designed DRs when only the simplest problem features are used as terminal nodes. This shows how important it is to design appropriate composite terminal nodes to produce high-quality DRs.

Heuristic Ensemble Construction Methods of Automatically Designed Dispatching Rules for the Unrelated Machines Environment

Dynamic scheduling represents an important class of combinatorial optimisation problems that are usually solved with simple heuristics, the so-called dispatching rules (DRs). Designing efficient DRs is a tedious task, which is why it has been automated through the application of genetic programming (GP). Various approaches have been used to improve the results of automatically generated DRs, with ensemble learning being one of the best-known. The goal of ensemble learning is to create sets of automatically designed DRs that perform better together. One of the main problems in ensemble learning is the selection of DRs to form the ensemble. To this end, various ensemble construction methods have been proposed over the years. However, these methods are quite computationally intensive and require a lot of computation time to obtain good ensembles. Therefore, in this study, we propose several simple heuristic ensemble construction methods that can be used to construct ensembles quite efficiently and without the need to evaluate their performance. The proposed methods construct the ensembles solely based on certain properties of the individual DRs used for their construction. The experimental study shows that some of the proposed heuristic construction methods perform better than more complex state-of-the-art approaches for constructing ensembles.

Minimizing total completion time with machine-dependent priority lists

We consider a natural, yet challenging variant of the parallel machine scheduling problem in which each machine imposes a preferential order over the jobs and schedules the jobs accordingly once assigned to it. We study the problem of minimizing the total completion time, distinguishing between identical and unrelated machines, machine-dependent and identical priority lists, or a constant number of different priority classes. Additionally, we consider the setting in which the priority list on a machine must satisfy longest processing time first. We resolve the computational complexity of the problem and provide a clear distinction between problems that are polynomial time solvable and APX-hard.

Weighted tardiness minimisation for unrelated machines with sequence-dependent and resource-constrained setups

Motivated by the need of quick job (re-)scheduling, we examine an elaborate scheduling environment under the objective of total weighted tardiness minimisation. The examined problem variant moves well beyond existing literature, as it considers unrelated machines, sequence-dependent and machine-dependent setup times, and a renewable resource constraint on the number of simultaneous setups. For this variant, we provide a relaxed MILP to calculate lower bounds, thus estimating a worst-case optimality gap. As a fast exact approach appears not plausible for instances of practical importance, we extend known (meta-)heuristics to deal with the problem at hand, coupling them with a Constraint Programming (CP) component – vital to guarantee the non-violation of the problem's constraints – which optimally allocates resources with respect to tardiness minimisation. The validity and versatility of employing different (meta-)heuristics exploiting a relaxed MILP as a quality measure are revealed by our extensive experimental study, which shows that the methods deployed have complementary strengths depending on the instance parameters. Since the problem description has been obtained from a textile manufacturer where jobs of diverse size arrive continuously under tight due dates, we also discuss the practical impact of our approach in terms of both tardiness decrease and broader managerial insights.

Combining single objective dispatching rules into multi-objective ensembles for the dynamic unrelated machines environment

Combining single objective dispatching rules into multi-objective ensembles for the dynamic unrelated machines environment

A Proof of the Nisan-Ronen Conjecture --- An Overview

This note presents an overview of our recent publication, which validates a conjecture proposed by Nisan and Ronen in their seminal paper [Nisan and Ronen 2001]. We show that the optimal approximation ratio for deterministic truthful mechanisms for makespan-minimization by a set of n unrelated machines is n.

Review on unrelated parallel machine scheduling problem with additional resources

This study deals with an unrelated parallel machine scheduling problem with additional resources (UPMR). That is one of the important sub-problems in the scheduling. UPMR consists of scheduling a set of jobs on unrelated machines. In addition to that, a number of one or more additional resources are needed. UPMR is very important and its importance comes from the wealth of applications; they are applicable to engineering and scientific situations and manufacturing systems such as industrial robots, nurses, machine operators, bus drivers, tools, assembly plant machines, fixtures, pallets, electricity, mechanics, dies, automated guided vehicles, fuel, and more. The importance also comes from the concern about the limitation of resources that are dedicated for the production process. Therefore, researchers and decision makers are still working on UPMR problem to get an optimum schedule for all instances which have not been obtained to this day. The optimum schedule is able to increase the profits and decrease the costs whilst satisfying the customers’ needs. This research aims to review and discuss studies related to unrelated parallel machines and additional resources. Overall, the review demonstrates the criticality of resolving the UPMR problem. Metaheuristic techniques exhibit significant effectiveness in generating results and surpassing other algorithms. Nevertheless, continued improvement is essential to satisfy the evolving requirements of UPMR, which are subject to operational changes based on customer demand.

Review on unrelated parallel machine scheduling problem with additional resources

This study deals with an unrelated parallel machine scheduling problem with additional resources (UPMR). That is one of the important sub-problems in the scheduling. UPMR consists of scheduling a set of jobs on unrelated machines. In addition to that, a number of one or more additional resources are needed. UPMR is very important and its importance comes from the wealth of applications; they are applicable to engineering and scientific situations and manufacturing systems such as industrial robots, nurses, machine operators, bus drivers, tools, assembly plant machines, fixtures, pallets, electricity, mechanics, dies, automated guided vehicles, fuel, and more. The importance also comes from the concern about the limitation of resources that are dedicated for the production process. Therefore, researchers and decision makers are still working on UPMR problem to get an optimum schedule for all instances which have not been obtained to this day. The optimum schedule is able to increase the profits and decrease the costs whilst satisfying the customers’ needs. This research aims to review and discuss studies related to unrelated parallel machines and additional resources. Overall, the review demonstrates the criticality of resolving the UPMR problem. Metaheuristic techniques exhibit significant effectiveness in generating results and surpassing other algorithms. Nevertheless, continued improvement is essential to satisfy the evolving requirements of UPMR, which are subject to operational changes based on customer demand.

Improving Additive Manufacturing production planning: A sub-second pixel-based packing algorithm

Improving Additive Manufacturing production planning: A sub-second pixel-based packing algorithm

Constructing ensembles of dispatching rules for multi-objective tasks in the unrelated machines environment

Scheduling is a frequently studied combinatorial optimisation problem that often needs to be solved under dynamic conditions and to optimise multiple criteria. The most commonly used method for solving dynamic problems are dispatching rules (DRs), simple constructive heuristics that build the schedule incrementally. Since it is difficult to design DRs manually, they are often created automatically using genetic programming. Although such rules work well, their performance is still limited and various methods, especially ensemble learning, are used to improve them. So far, ensembles have only been used in the context of single-objective scheduling problems. This study aims to investigate the possibility of constructing ensembles of DRs for solving multi-objective (MO) scheduling problems. To this end, an existing ensemble construction method called SEC is adapted by extending it with non-dominated sorting to construct Pareto fronts of ensembles for a given MO problem. In addition, the algorithms NSGA-II and NSGA-III were adapted to construct ensembles and compared with the SEC method to demonstrate their effectiveness. All methods were evaluated on four MO problems with different number of criteria to be optimised. The results show that ensembles of DRs achieve better Pareto fronts compared to individual DRs. Moreover, the results show that SEC achieves equally good or even slightly better results than NSGA-II and NSGA-III when constructing ensembles, while it is simpler and slightly less computationally expensive. This shows the potential of using ensembles to increase the performance of individual DRs for MO problems.