Enhanced energetic reasoning-based lower bounds for the resource constrained project scheduling problem

We present new and effective lower bounds for the resource constrained project scheduling problem. This problem is widely known to be notoriously difficult to solve due to the lack of lower bounds that are both tight and fast. In this paper, we propose several new lower bounds that are based on the concept of energetic reasoning. A major contribution of this work is to investigate several enhanced new feasibility tests that prove useful for deriving new lower bounds that consistently outperform the classical energetic reasoning-based lower bound. In particular, we present the results of a comprehensive computational study, carried out on 1560 benchmark instances, that provides strong evidence that a deceptively simple dual feasible function-based lower bound is highly competitive with a state-of-the-art lower bound while being extremely fast. Furthermore, we found that an effective shaving procedure enables to derive an excellent lower bound that often outperforms the best bound from the literature while being significantly simpler.     

Branch‐and‐bound approach for optima localization in scheduling multiprocessor jobs

We consider multiprocessor task scheduling problems with dedicated processors. We determine the tight optima localization intervals for different subproblems of the basic problem. Based on the ideas of a computer‐aided technique developed by Sevastianov and Tchernykh for shop scheduling problems, we elaborate a similar method for the multiprocessor task scheduling problem. Our method allows us to find an upper bound for the length of the optimal schedule in terms of natural lower bound. As a byproduct of our results, a family of linear‐time approximation algorithms with a constant ratio performance guarantee is designed for the NP‐hard subproblems of the basic problem, and new polynomially solvable classes of problems are found.     

Using column generation to solve parallel machine scheduling problems with minmax objective functions

In this paper we consider the parallel machine scheduling problem of minimizing an objective function of the minmax type, like maximum lateness, subject to release dates, deadlines, and/or generalized precedence constraints. We use a destructive strategy to compute a lower bound. Here we test the feasibility of a decision problem by applying column generation to compute a bound on the number of machines that we need to feasibly accommodate all jobs. After having derived the lower bound, we try to find a matching upper bound by identifying a feasible schedule with objective function value equal to this lower bound. Our computational results show that our lower bound is so strong that this is almost always possible. We are able to solve problems with up to 160 jobs and 10 machines in 10 minutes on average.     

A branch-and-bound algorithm for single machine scheduling with quadratic earliness and tardiness penalties

This paper considers the problem of scheduling a single machine, in which the objective function is to minimize the weighted quadratic earliness and tardiness penalties and no machine idle time is allowed. We develop a branch and bound algorithm involving the implementation of lower and upper bounding procedures as well as some dominance rules. The lower bound is designed based on a lagrangian relaxation method and the upper bound includes two phases, one for constructing initial schedules and the other for improving them. Computational experiments on a set of randomly generated instances show that one of the proposed heuristics, used as an upper bound, has an average gap less than 1.3% for instances optimally solved. The results indicate that both the lower and upper bounds are very tight and the branch-and-bound algorithm is the first algorithm that is able to optimally solve problems with up to 30 jobs in a reasonable amount of time.     

Stochastic and semidefinite optimization for scheduling in orthogonal frequency division multiple access networks

In this paper, we propose stochastic binary quadratic programs for the scheduling resource allocation process of a wireless orthogonal frequency division multiple access network. More precisely, we formulate a two-stage stochastic model, then we further extend the two-stage model by introducing a knapsack probabilistic constrained approach, and finally we propose a multi-stage stochastic program for this problem. The models are aimed at minimizing the total power consumption of the network at each time slot of the scheduling process subject to user bit rates, sub-carrier and modulation linear constraints. In order to compute lower bounds, we derive linear and semidefinite programming relaxations for each of the proposed models. The bounds are also compared with a basic variable neighborhood search metaheuristic approach. Numerical results show tight lower bounds for the semidefinite relaxations when compared to the linear ones and with the metaheuristic. Moreover, near optimal solutions are found with the semidefinite relaxations for the two-stage model without using probabilistic constraints and for the multi-stage program as well.     

COMPUTATION OF BEST BOUNDS OF PROBABILITIES FROM UNCERTAIN DATA

An uncertainty reasoning method is presented in this article. The method can be used to compute from a given set of conditional probabilities the best lower bounds and the best upper bounds of those conditional probabilities that are not explicitly provided. The computation of the best upper(lower) bound of such a conditional probability relies on solution of a linear programming problem. Some reduction techniques are proposed in this article to improve the efficiency of our uncertainty reasoning method. As illustrated in Section 4.3, for many uncertainty reasoning problems in medical diagnosis, by using our reduction techniques, the best range of a conditional probability, which is specified by a lower bound and an upper bound, can be computed in polynomial time in terms of the number of basic events involved in the reasoning.     

Semi-online scheduling problems on a small number of machines

We consider the semi-online parallel machine scheduling problem of minimizing the makespan given a priori information: the total processing time, the largest processing time, the combination of the previous two or the optimal makespan. We propose a new algorithm that can be applied to the problem with the known total or largest processing time and prove that it has improved competitive ratios for the cases with a small number of machines. Improved lower bounds of the competitive ratio are also provided by presenting adversary lower bound examples.     

Scheduling Parallel Machines for the Customer Order Problem

This paper considers scheduling problems where jobs are dispatched in batches. The objective is to minimize the sum of the completion times of the batches. While a machine can process only one job at a time, multiple machines can simultaneously process jobs in a batch. This simple environment has a variety of real world applications such as part kitting and customer order scheduling.A heuristic is presented for the parallel machine version of the problem. Also, a tight worst case bound on the relative error is found. For the case of two parallel machines, we examine two heuristics, which are based on simple scheduling rules. We find tight worst case bounds of 6/5 and 9/7 on the relative error and show that neither procedure is superior for all instances. Finally, we empirically evaluate these two heuristics. For large problems, the methods find solutions that are close to optimal.     

A tight lower bound for job scheduling with cancellation

The Job Scheduling with Cancellation problem is a variation of classical scheduling problems in which jobs can be cancelled while waiting for execution. In this paper we prove a tight lower bound of 5 for the competitive ratio of any deterministic online algorithm for this problem, for the case where all jobs have the same processing time.     

Infra-chromatic bound for exact maximum clique search

Many efficient exact branch and bound maximum clique solvers use approximate coloring to compute an upper bound on the clique number for every subproblem. This technique reasonably promises tight bounds on average, but never tighter than the chromatic number of the graph.Li and Quan, 2010, AAAI Conference, p. 128–133 describe a way to compute even tighter bounds by reducing each colored subproblem to maximum satisfiability problem (MaxSAT). Moreover they show empirically that the new bounds obtained may be lower than the chromatic number.Based on this idea this paper shows an efficient way to compute related “infra-chromatic” upper bounds without an explicit MaxSAT encoding. The reported results show some of the best times for a stand-alone computer over a number of instances from standard benchmarks.     

A branch and bound algorithm for an identical parallel machine scheduling problem with a job splitting property

We focus on the problem of scheduling n independent jobs on m identical parallel machines with the objective of minimizing total tardiness of the jobs considering a job splitting property. In this problem, it is assumed that a job can be split into sub-jobs and these sub-jobs can be processed independently on parallel machines. We develop several dominance properties and lower bounds for the problem, and suggest a branch and bound algorithm using them. Computational experiments are performed on randomly generated test problems and results show that the suggested algorithm solves problems of moderate sizes in a reasonable amount of computation time.     

Lower bounds for maximal and convex layers problems

In this paper we derive tight lower bounds for the maximal and convex layers problems in the plane. Our lower bound proofs for the maxima problem and convex hull problem are simpler than those previously known. We also obtain an Ω(nlog n) lower bound for the maximal depth problem, and the convex depth problem, when the points are given in sorted order of their x-coordinates.     

Scheduling of coupled tasks and one-machine no-wait robotic cells

Coupled task scheduling problems have been known for more than 25 years. Several complexity results have been established in the meantime, but the status of the identical task case remains still unsettled. We describe a new class of equivalent one-machine no-wait robotic cell problems. It turns out that scheduling of identical coupled tasks corresponds to the production of a single part type in the robotic cell. We shall describe new algorithmic procedures to solve this robotic cell problem, allowing lower and upper bounds on the production time and discussing in particular cyclic production plans.     

Minimizing Makespan and Preemption Costs on a System of Uniform Machines

It is well known that for preemptive scheduling on uniform machines there exist polynomial time exact algorithms, whereas for non-preemptive scheduling there are probably no such algorithms. However, it is not clear how many preemptions (in total, or per job) suffice in order to guarantee an optimal polynomial time algorithm. In this paper we investigate exactly this hardness gap, formalized as two variants of the classic preemptive scheduling problem. In generalized multiprocessor scheduling (GMS) we have a job-wise or total bound on the number of preemptions throughout a feasible schedule. We need to find a schedule that satisfies the preemption constraints, such that the maximum job completion time is minimized. In minimum preemptions scheduling (MPS) the only feasible schedules are preemptive schedules with the smallest possible makespan. The goal is to find a feasible schedule that minimizes the overall number of preemptions. Both problems are NP-hard, even for two machines and zero preemptions. For GMS, we develop polynomial time approximation schemes, distinguishing between the cases where the number of machines is fixed, or given as part of the input. Our scheme for a fixed number of machines has linear running time, and can be applied also for instances where jobs have release dates, and for instances with arbitrary preemption costs. For MPS, we derive matching lower and upper bounds on the number of preemptions required by any optimal schedule. Our results for MPS hold for any instance in which a job, J_j, can be processed simultaneously by ρ_j machines, for some ρ_j ≥ 1.     

A branch and bound algorithm for minimizing total completion time on a single batch machine with incompatible job families and dynamic arrivals

In this paper, we consider a single batch machine scheduling problem with incompatible job families and dynamic job arrivals. The objective is to minimize the total completion time. This problem is known to be strongly NP-hard. We present several dominance properties and two types of lower bounds, which are incorporated to construct a basic branch and bound algorithm. Furthermore, according to the characteristics of dynamic job arrivals, a decomposed branch and bound algorithm is proposed to improve the efficiency. The proposed algorithms are tested on a large set of randomly generated problem instances.     

Lower bounds for minimizing total completion time in a two-machine flow shop

For the -hard problem of scheduling n jobs in a two-machine flow shop so as to minimize the total completion time, we present two equivalent lower bounds that are computable in polynomial time. We formulate the problem by the use of positional completion time variables, which results in two integer linear programming formulations with O(n ²) variables and O(n) constraints. Solving the linear programming relaxation renders a very strong lower bound with an average relative gap of only 0.8% for instances with more than 30 jobs. We further show that relaxing the formulation in terms of positional completion times by applying Lagrangean relaxation yields the same bound, no matter which set of constraints we relax.     

Cone-bounded nonlinearities and mean-square bounds—Smoothing and prediction

Mean-square performance bounds are derived for smoothing and prediction problems associated with the broad class of nonlinear dynamic systems which, when modeled by Ito differential equations, contain drift (·dt) coefficients which are, to within a uniformly Lipschitz residual, jointly linear in the system state and externally applied control. Included in this paper are lower bounds on the error covariance attainable by any smoother or any predictor, including the optimum, and upper bounds on the performance of some simple, implementable predictors reminiscent of the designs which are optimal in the linear case. The lower bounds on smoothing and prediction performance are established using measure-transformation techniques to relate a version of the nonlinear problem to its linearization. The upper bound on prediction performance is constructed by a direct analysis of the estimation error. All the bounds hold for correlated system and observation noises. All are rigorously derived and independent of control or control law. In each case, the computational effort is comparable to that for the corresponding optimum linear smoothing or prediction problem. The bounds converge with vanishing nonlinearity (vanishing Lipschitz constants) to the known optimum performance for the limiting linear system. Consequently, the bounds are asymptotically tight and the simple designs studied are asymptotically optimal with vanishing nonlinearity.     

Online algorithms for scheduling two parallel machines with a single server

We consider an online scheduling problem on two identical parallel machines with a single server. Jobs arrive one by one and each job has to be loaded by the server before being processed on one of the machines, and unloaded immediately by the server after its processing. Both loading and unloading times are equal to one time unit. The goal is to minimize the makespan. For the variant of the problem involving both loading and unloading operations, we present an online algorithm with competitive ratio of 5/3. For the variant with loading operation only, we show that the competitive ratio of list scheduling is at least 8/5 and provide an improved online algorithm with competitive ratio of 11/7. Finally, we discuss the lower bounds for these problems. We show that both variants have a lower bound of 3/2. Furthermore, we show that the lower bound of the first variant is at least 8/5 if the online algorithm satisfies a certain constraint.     

Sensitive Functions and Approximate Problems

Properties of functions that are good measures of the CRCW PRAM complexity of computing them are investigated. While theblock sensitivityis known to be a good measure of the CREW PRAM complexity, no such measure is known for CRCW PRAMs. It is shown that the complexity of computing a function is related to itseverywhere sensitivity, introduced by Vishkin and Wigderson. Specifically, the time required to compute a functionf: Dⁿ→Rof everywhere sensitivityes(f) withPprocessors and unbounded memory isΩ(log[log es(f)/(log(|D|+4P/es(f)))]). This improves results of Azar and of Vishkin and Wigderson. This lower bound is used to derive new lower bounds for someapproximate problems. These problems can often be solved faster than their exact counterparts and for many applications, it is sufficient to solve the approximate problem. It is shown thatapproximate selection,approximate counting,approximate compaction, andpadded sortingall require timeΩ(log log n) with a linear number of processors, if the level of accuracy desired is moderately high. For these levels of accuracy, no lower bounds were known for these problems on the PRAM model. The lower bounds for some of the problems are tight.