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Single-machine Scheduling
Single-machine scheduling or single-resource scheduling is an optimization problem in computer science and operations research. We are given ''n'' jobs ''J''1, ''J''2, ..., ''Jn'' of varying processing times, which need to be scheduled on a single machine, in a way that optimizes a certain objective, such as the throughput. Single-machine scheduling is a special case of identical-machines scheduling, which is itself a special case of optimal job scheduling. Many problems, which are NP-hard in general, can be solved in polynomial time in the single-machine case. In the standard three-field notation for optimal job scheduling problems, the single-machine variant is denoted by 1 in the first field. For example, " 1, , \sum C_j" is a single-machine scheduling problem with no constraints, where the goal is to minimize the sum of completion times. The makespan-minimization problem 1, , C_, which is a common objective with multiple machines, is trivial with a single machine, since the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Problem
In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ..., an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: * An optimization problem with discrete variables is known as a '' discrete optimization'', in which an object such as an integer, permutation or graph must be found from a countable set. * A problem with continuous variables is known as a '' continuous optimization'', in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems. Search space In the context of an optim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly NP-hard
In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters. A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input. A problem is said to be strongly NP-hard if a strongly NP-complete problem has a pseudo-polynomial reduction to it. This pseudo-polynomial reduction is more restrictive than the usual poly-time reduction used for NP-hardness proofs. In special, the pseudo-polynomial reduction cannot output a numerical parameter that is not polinomially bounded by the size and value of numbers in the input ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tabu Search
Tabu search (TS) is a metaheuristic search method employing local search methods used for mathematical optimization. It was created by Fred W. Glover in 1986 and formalized in 1989. Local (neighborhood) searches take a potential solution to a problem and check its immediate neighbors (that is, solutions that are similar except for very few minor details) in the hope of finding an improved solution. Local search methods have a tendency to become stuck in suboptimal regions or on plateaus where many solutions are equally fit. Tabu search enhances the performance of local search by relaxing its basic rule. First, at each step ''worsening'' moves can be accepted if no improving move is available (like when the search is stuck at a strict local minimum). In addition, ''prohibitions'' (hence the term ''tabu'') are introduced to discourage the search from coming back to previously-visited solutions. The implementation of tabu search uses memory structures that describe the visited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ant Colony Optimization
In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems that can be reduced to finding good paths through graphs. Artificial ants represent multi-agent methods inspired by the behavior of real ants. The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of artificial ants and local search algorithms have become a preferred method for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. As an example, ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones to direct each other to resources while exploring their environment. The simulated 'ants' similarly record their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simulated Annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. For large numbers of local optima, SA can find the global optimum. It is often used when the search space is discrete (for example the traveling salesman problem, the boolean satisfiability problem, protein structure prediction, and job-shop scheduling). For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be preferable to exact algorithms such as gradient descent or branch and bound. The name of the algorithm comes from annealing in metallurgy, a technique involving heating and controlled cooling of a material to alter its physical properties. Both are attributes of the material that depend on their thermodynamic free energy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neural Network
A neural network is a group of interconnected units called neurons that send signals to one another. Neurons can be either biological cells or signal pathways. While individual neurons are simple, many of them together in a network can perform complex tasks. There are two main types of neural networks. *In neuroscience, a '' biological neural network'' is a physical structure found in brains and complex nervous systems – a population of nerve cells connected by synapses. *In machine learning, an '' artificial neural network'' is a mathematical model used to approximate nonlinear functions. Artificial neural networks are used to solve artificial intelligence problems. In biology In the context of biology, a neural network is a population of biological neurons chemically connected to each other by synapses. A given neuron can be connected to hundreds of thousands of synapses. Each neuron sends and receives electrochemical signals called action potentials to its conne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genetic Algorithm
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to generate high-quality solutions to optimization and search problems via biologically inspired operators such as selection, crossover, and mutation. Some examples of GA applications include optimizing decision trees for better performance, solving sudoku puzzles, hyperparameter optimization, and causal inference. Methodology Optimization problems In a genetic algorithm, a population of candidate solutions (called individuals, creatures, organisms, or phenotypes) to an optimization problem is evolved toward better solutions. Each candidate solution has a set of properties (its chromosomes or genotype) which can be mutated and altered; traditionally, solutions are represented in binary as strings of 0s and 1s, but other encod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Scheduling
Interval scheduling is a class of problems in computer science, particularly in the area of algorithm design. The problems consider a set of tasks. Each task is represented by an ''interval'' describing the time in which it needs to be processed by some machine (or, equivalently, scheduled on some resource). For instance, task A might run from 2:00 to 5:00, task B might run from 4:00 to 10:00 and task C might run from 9:00 to 11:00. A subset of intervals is ''compatible'' if no two intervals overlap on the machine/resource. For example, the subset is compatible, as is the subset ; but neither nor are compatible subsets, because the corresponding intervals within each subset overlap. The ''interval scheduling maximization problem'' (ISMP) is to find a largest compatible set, i.e., a set of non-overlapping intervals of maximum size. The goal here is to execute as many tasks as possible, that is, to maximize the throughput. It is equivalent to finding a Independent set (graph theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Problem
In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset ''S'' of positive integers can be partition of a set, partitioned into two subsets ''S''1 and ''S''2 such that the sum of the numbers in ''S''1 equals the sum of the numbers in ''S''2. Although the partition problem is NP-complete, there is a pseudo-polynomial time dynamic programming solution, and there are Heuristic, heuristics that solve the problem in many instances, either optimally or approximately. For this reason, it has been called "the easiest hard problem". There is an optimization problem, optimization version of the partition problem, which is to partition the multiset ''S'' into two subsets ''S''1, ''S''2 such that the difference between the sum of elements in ''S''1 and the sum of elements in ''S''2 is minimized. The optimization version is NP-hard, but can be solved efficiently in practice. The partition problem is a special case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-partition
The 3-partition problem is a strongly NP-complete problem in computer science. The problem is to decide whether a given multiset of integers can be partitioned into triplets that all have the same sum. More precisely: * Input: a multiset ''S'' containing ''n'' positive integer elements. * Conditions: ''S'' must be partitionable into ''m'' triplets, ''S''1, ''S''2, …, ''S''''m'', where ''n = 3m''. These triplets partition ''S'' in the sense that they are disjoint and they cover ''S''. The target value ''T'' is computed by taking the sum of all elements in ''S'', then dividing by ''m''. * Output: whether or not there exists a partition of ''S'' such that, for all triplets, the sum of the elements in each triplet equals ''T''. The 3-partition problem remains strongly NP-complete under the restriction that every integer in ''S'' is strictly between ''T''/4 and ''T''/2. Example # The set S = \ can be partitioned into the four sets \, \, \ , \, each of which sums to ''T'' = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Scheduling
Interval scheduling is a class of problems in computer science, particularly in the area of algorithm design. The problems consider a set of tasks. Each task is represented by an ''interval'' describing the time in which it needs to be processed by some machine (or, equivalently, scheduled on some resource). For instance, task A might run from 2:00 to 5:00, task B might run from 4:00 to 10:00 and task C might run from 9:00 to 11:00. A subset of intervals is ''compatible'' if no two intervals overlap on the machine/resource. For example, the subset is compatible, as is the subset ; but neither nor are compatible subsets, because the corresponding intervals within each subset overlap. The ''interval scheduling maximization problem'' (ISMP) is to find a largest compatible set, i.e., a set of non-overlapping intervals of maximum size. The goal here is to execute as many tasks as possible, that is, to maximize the throughput. It is equivalent to finding a Independent set (graph theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
