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Unrelated-machines Scheduling
Unrelated-machines scheduling is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. We need to schedule ''n'' jobs ''J''1, ''J''2, ..., ''Jn'' on ''m'' different machines, such that a certain objective function is optimized (usually, the makespan should be minimized). The time that machine ''i'' needs in order to process job j is denoted by ''pi,j''. The term ''unrelated'' emphasizes that there is no relation between values of ''pi,j'' for different ''i'' and ''j''. This is in contrast to two special cases of this problem: uniform-machines scheduling - in which ''pi,j'' = ''pi'' / ''sj'' (where ''sj'' is the speed of machine ''j''), and identical-machines scheduling - in which ''pi,j'' = ''pi'' (the same run-time on all machines). In the standard three-field notation for optimal job scheduling problems, the unrelated-machines variant is denoted by R in the first field. For example, the problem denoted by " R, , C_\max" ... [...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] |
FPTAS
A fully polynomial-time approximation scheme (FPTAS) is an algorithm for finding approximate solutions to function problems, especially optimization problems. An FPTAS takes as input an instance of the problem and a parameter ε > 0. It returns as output a value which is at least 1-\varepsilon times the correct value, and at most 1 + \varepsilon times the correct value. In the context of optimization problems, the correct value is understood to be the value of the optimal solution, and it is often implied that an FPTAS should produce a valid solution (and not just the value of the solution). Returning a value and finding a solution with that value are equivalent assuming that the problem possesses self reducibility. Importantly, the run-time of an FPTAS is polynomial in the problem size and in 1/ε. This is in contrast to a general polynomial-time approximation scheme (PTAS). The run-time of a general PTAS is polynomial in the problem size for each specific ε, but might ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithmic Mechanism Design
Algorithmic mechanism design (AMD) lies at the intersection of economic game theory, optimization, and computer science. The prototypical problem in mechanism design is to design a system for multiple self-interested participants, such that the participants' self-interested actions at equilibrium lead to good system performance. Typical objectives studied include revenue maximization and social welfare maximization. Algorithmic mechanism design differs from classical economic mechanism design in several respects. It typically employs the analytic tools of theoretical computer science, such as worst case analysis and approximation ratios, in contrast to classical mechanism design in economics which often makes distributional assumptions about the agents. It also considers computational constraints to be of central importance: mechanisms that cannot be efficiently implemented in polynomial time are not considered to be viable solutions to a mechanism design problem. This often, for exa ... [...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] |
Configuration Linear Program
The configuration linear program (configuration-LP) is a linear programming technique used for solving combinatorial optimization problems. It was introduced in the context of the cutting stock problem. Later, it has been applied to the bin packing and job scheduling problems. In the configuration-LP, there is a variable for each possible ''configuration'' - each possible multiset of items that can fit in a single bin (these configurations are also known as ''patterns'') . Usually, the number of configurations is exponential in the problem size, but in some cases it is possible to attain approximate solutions using only a polynomial number of configurations. In bin packing The integral LP In the bin packing problem, there are ''n'' items with different sizes. The goal is to pack the items into a minimum number of bins, where each bin can contain at most ''B''. A ''feasible configuration'' is a set of sizes with a sum of at most ''B''. * ''Example'': suppose the item sizes ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max-min Item Allocation
Egalitarian item allocation, also called max-min item allocation is a fair item allocation problem, in which the fairness criterion follows the egalitarian social choice rule, egalitarian rule. The goal is to maximize the minimum value of an agent. That is, among all possible allocations, the goal is to find an allocation in which the smallest value of an agent is as large as possible. In case there are two or more allocations with the same smallest value, then the goal is to select, from among these allocations, the one in which the second-smallest value is as large as possible, and so on (by the leximin order). Therefore, an egalitarian item allocation is sometimes called a leximin item allocation. The special case in which the value of each item ''j'' to each agent is either 0 or some constant ''vj'' is called the santa claus problem: santa claus has a fixed set of gifts, and wants to allocate them among children such that the least-happy child is as happy as possible. Some r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomized Rounding
In computer science and operations research, randomized rounding is a widely used approach for designing and analyzing approximation algorithms. Many combinatorial optimization problems are computationally intractability (complexity), intractable to solve exactly (to optimality). For such problems, randomized rounding can be used to design fast (polynomial time) approximation algorithms—that is, algorithms that are guaranteed to return an approximately optimal solution given any input. The basic idea of randomized rounding is to convert an optimal solution of a linear programming relaxation, relaxation of the problem into an approximately-optimal solution to the original problem. The resulting algorithm is usually analyzed using the probabilistic method. Overview The basic approach has three steps: # Formulate the problem to be solved as an integer linear program (ILP). # Compute an optimal fractional solution x to the linear programming relaxation (LP) of the ILP. # Rou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knapsack Problem
The knapsack problem is the following problem in combinatorial optimization: :''Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.'' It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision-makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively. The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. The subset sum problem is a special case of the decision and 0-1 problems where each kind of item, the weight equals the value: w_i=v_i. In the field of cryptography, the term ''knapsack problem'' is often used to refer specifically to the subset sum pro ... [...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] |
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] |
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] |
Local Search (optimization)
In computer science, local search is a heuristic method for solving computationally hard Mathematical optimization, optimization problems. Local search can be used on problems that can be formulated as finding a solution that maximizes a criterion among a number of candidate solutions. Local Search algorithm, search algorithms move from solution to solution in the space of candidate solutions (the ''search space'') by applying local changes, until a solution deemed optimal is found or a time bound is elapsed. Local search algorithms are widely applied to numerous hard computational problems, including problems from computer science (particularly artificial intelligence), mathematics, operations research, engineering, and bioinformatics. Examples of local search algorithms are WalkSAT, the 2-opt, 2-opt algorithm for the Traveling Salesman Problem and the Metropolis–Hastings algorithm. While it is sometimes possible to substitute gradient descent for a local search algorithm, gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
