A job allocation algorithm for parallel processors

We are given a nite set of jobs of equal processing times with readiness times and tails and a set of identical processors. The aim is to schedule the given set of jobs on the given set of processors to minimize the total processing time (or makespan). An algorithm for that problem with the time complexity O(n logn) was proposed earlier in [10]. This algorithm improves the running time of the previously known best algorithm [9] under the assumption that the tails of all jobs are bounded by some constant. In this paper we show that an algorithm based on the ideas of the algorithm from [10] can be constructed in which the above restriction is removed.Eje: Procesamiento distribuido y paralelo. Tratamiento de señalesRed de Universidades con Carreras en Informática (RedUNCI

A job allocation algorithm for parallel processors

We are given a nite set of jobs of equal processing times with readiness times and tails and a set of identical processors. The aim is to schedule the given set of jobs on the given set of processors to minimize the total processing time (or makespan). An algorithm for that problem with the time complexity O(n logn) was proposed earlier in [10]. This algorithm improves the running time of the previously known best algorithm [9] under the assumption that the tails of all jobs are bounded by some constant. In this paper we show that an algorithm based on the ideas of the algorithm from [10] can be constructed in which the above restriction is removed.Eje: Procesamiento distribuido y paralelo. Tratamiento de señalesRed de Universidades con Carreras en Informática (RedUNCI

Variable Parameter Analysis for Scheduling One Machine

In contrast to the fixed parameter analysis (FPA), in the variable parameter analysis (VPA) the value of the target problem parameter is not fixed, it rather depends on the structure of a given problem instance and tends to have a favorable asymptotic behavior when the size of the input increases. While applying the VPA to an intractable optimization problem with $n$ objects, the exponential-time dependence in enumeration of the feasible solution set is attributed solely to the variable parameter $\nu$, $\nu<<n$. As opposed to the FPA, the VPA does not imply any restriction on some problem parameters, it rather takes an advantage of a favorable nature of the problem, which permits to reduce the cost of enumeration of the solution space. Our main technical contribution is a variable parameter algorithm for a strongly $\mathsf{NP}$-hard single-machine scheduling problem to minimize maximum job lateness. The target variable parameter $\nu$ is the number of jobs with some specific characteristics, the ``emerging'' ones. The solution process is separated in two phases. At phase 1 a partial solution including $n-\nu$ non-emerging jobs is constructed in a low degree polynomial time. At phase 2 less than $\nu!$ permutations of the $\nu$ emerging jobs are considered. Each of them are incorporated into the partial schedule of phase 1. Doe to the results of an earlier conducted experimental study, $\nu/n$ varied from $1/4$ for small problem instances to $1/10$ for the largest tested problem instances, so that that the ratio becomes closer to 0 for large $n$s.Comment: arXiv admin note: substantial text overlap with arXiv:2103.0990

Efficient Heuristics for Scheduling with Release and Delivery Times

In this chapter, we describe efficient heuristics for scheduling jobs with release and delivery times with the objective to minimize the maximum job completion time. These heuristics are essentially based on a commonly used scheduling theory in Jackson’s extended heuristic. We present basic structural properties of the solutions delivered by Jackson’s heuristic and then illustrate how one can exploit them to build efficient heuristics

A Brief Look at Multi-Criteria Problems: Multi-Threshold Optimization versus Pareto-Optimization

Multi-objective optimization problems are important as they arise in many practical circumstances. In such problems, there is no general notion of optimality, as there are different objective criteria which can be contradictory. In practice, often there is no unique optimality criterion for measuring the solution quality. The latter is rather determined by the value of the solution for each objective criterion. In fact, a practitioner seeks for a solution that has an acceptable value of each of the objective functions and, in practice, there may be different tolerances to the quality of the delivered solution for different objective functions: for some objective criteria, solutions that are far away from an optimal one can be acceptable. Traditional Pareto-optimality approach aims to create all non-dominated feasible solutions in respect to all the optimality criteria. This often requires an inadmissible time. Besides, it is not evident how to choose an appropriate solution from the Pareto-optimal set of feasible solutions, which can be very large. Here we propose a new approach and call it multi-threshold optimization setting that takes into account different requirements for different objective criteria and so is more flexible and can often be solved in a more efficient way

A job allocation algorithm for parallel processors

We are given a nite set of jobs of equal processing times with readiness times and tails and a set of identical processors. The aim is to schedule the given set of jobs on the given set of processors to minimize the total processing time (or makespan). An algorithm for that problem with the time complexity O(n logn) was proposed earlier in [10]. This algorithm improves the running time of the previously known best algorithm [9] under the assumption that the tails of all jobs are bounded by some constant. In this paper we show that an algorithm based on the ideas of the algorithm from [10] can be constructed in which the above restriction is removed.Eje: Procesamiento distribuido y paralelo. Tratamiento de señalesRed de Universidades con Carreras en Informática (RedUNCI

Preemptive Scheduling of Equal-Length Jobs to Maximize Weighted Throughput

We study the problem of computing a preemptive schedule of equal-length jobs with given release times, deadlines and weights. Our goal is to maximize the weighted throughput, which is the total weight of completed jobs. In Graham's notation this problem is described as (1 | r_j;p_j=p;pmtn | sum w_j U_j). We provide an O(n^4)-time algorithm for this problem, improving the previous bound of O(n^{10}) by Baptiste.Comment: gained one author and lost one degree in the complexit

An Efficient Heuristic for a Discrete Optimization Problem

In this paper we deal with a discrete optimization problem, which, among many other such problems, is computationally intractable. Since the existence of an exact solution algorithm for our problem is highly unlikely, the development of heuristic and approximation algorithms is of a great importance. Here we briefly discuss this issue and describe a robust 2-approximation heuristic that is used for getting an approximation solution for the problem of scheduling jobs with release times and due-dates on a single machine to minimize the maximum job lateness

Exact and Heuristic Algorithms for the Domination Problem

In a simple connected graph $G=(V,E)$, a subset of vertices $S \subseteq V$ is a dominating set if any vertex $v \in V\setminus S$ is adjacent to some vertex $x$ from this subset. A number of real-life problems can be modeled using this problem which is known to be among the difficult NP-hard problems in its class. We formulate the problem as an integer liner program (ILP) and compare the performance with the two earlier existing exact state-of-the-art algorithms and exact implicit enumeration and heuristic algorithms that we propose here. Our exact algorithm was able to find optimal solutions much faster than ILP and the above two exact algorithms for middle-dense instances. For graphs with a considerable size, our heuristic algorithm was much faster than both, ILP and our exact algorithm. It found an optimal solution for more than half of the tested instances, whereas it improved the earlier known state-of-the-art solutions for almost all the tested benchmark instances. Among the instances where the optimum was not found, it gave an average approximation error of $1.18$

f-polynomial on some graph operations

Given any function f:Z+→R+ , let us define the f-index If(G)=∑u∈V(G)f(du) and the f-polynomial Pf(G,x)=∑u∈V(G)x1/f(du)−1, for x>0 . In addition, we define Pf(G,0)=limx→0+Pf(G,x) . We use the f-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this f-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others.Supported in part by two grants from the Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI), and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain