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List Of Knapsack Problems
The knapsack problem is one of the most studied problems in combinatorial optimization, with many real-life applications. For this reason, many special cases and generalizations have been examined. Common to all versions are a set of ''n'' items, with each item 1 \leq j \leq n having an associated profit ''pj'' and weight ''wj''. The binary decision variable ''xj'' is used to select the item. The objective is to pick some of the items, with maximal total profit, while obeying that the maximum total weight of the chosen items must not exceed ''W''. Generally, these coefficients are scaled to become integers, and they are almost always assumed to be positive. The knapsack problem in its most basic form: __TOC__ Direct generalizations One common variant is that each item can be chosen multiple times. The bounded knapsack problem specifies, for each item ''j'', an upper bound ''uj'' (which may be a positive integer, or infinity) on the number of times item ''j'' can be selected: ... [...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] |
Quadratic Knapsack Problem
The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a weight, a value, and an extra profit that can be earned if two items are selected, determine the number of items to include in a collection without exceeding capacity of the knapsack, so as to maximize the overall profit. Usually, quadratic knapsack problems come with a restriction on the number of copies of each kind of item: either 0, or 1. This special type of QKP forms the 0-1 quadratic knapsack problem, which was first discussed by Gallo et al. The 0-1 quadratic knapsack problem is a variation of the knapsack problem, combining the features of the 0-1 knapsack problem and the quadratic knapsack problem. Definition Specifically, the 0–1 quadratic knapsack problem has the following form: : \text \left\ : \text X\equiv\left\. Here the binary variable ''xi'' represents whether ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Assignment Problem
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: :The problem instance has a number of ''agents'' and a number of ''tasks''. Any agent can be assigned to perform any task, incurring some ''cost'' that may vary depending on the agent-task assignment. It is required to perform as many tasks as possible by assigning at most one agent to each task and at most one task to each agent, in such a way that the ''total cost'' of the assignment is minimized. Alternatively, describing the problem using graph theory: :The assignment problem consists of finding, in a weighted graph, weighted bipartite graph, a Matching (graph theory), matching of maximum size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment, and the graph-theoretic version is called minimum-cost perfect matching. Otherwise, it is called unbalanced assig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cutting Stock Problem
In operations research, the cutting-stock problem is the problem of cutting standard-sized pieces of Inventory, stock material, such as paper rolls or sheet metal, into pieces of specified sizes while minimizing material wasted. It is an optimization (mathematics), optimization problem in mathematics that arises from applications in industry. In terms of Analysis of algorithms, computational complexity, the problem is an NP-hard problem reducible to the knapsack problem. The problem can be formulated as an integer linear programming problem. Illustration of one-dimensional cutting-stock problem A paper machine can produce an unlimited number of master (jumbo) rolls, each 5600 mm wide. The following 13 items must be cut, in the table below. The important thing about this kind of problem is that many different product units can be made from the same master roll, and the number of possible combinations is itself very large, in general, and not trivial to enumerate. The probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bin Packing Problem
The bin packing problem is an optimization problem, in which items of different sizes must be packed into a finite number of bins or containers, each of a fixed given capacity, in a way that minimizes the number of bins used. The problem has many applications, such as filling up containers, loading trucks with weight capacity constraints, creating file backups in media, splitting a network prefix into multiple subnets, and technology mapping in FPGA semiconductor chip design. Computationally, the problem is NP-hard, and the corresponding decision problem, deciding if items can fit into a specified number of bins, is NP-complete. Despite its worst-case hardness, optimal solutions to very large instances of the problem can be produced with sophisticated algorithms. In addition, many approximation algorithms exist. For example, the first fit algorithm provides a fast but often non-optimal solution, involving placing each item into the first bin in which it will fit. It requires ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-polynomial Time
In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the ''numeric value'' of the input (the largest integer present in the input)—but not necessarily in the ''length'' of the input (the number of bits required to represent it), which is the case for polynomial time algorithms.Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, 1979. In general, the numeric value of the input is exponential in the input length, which is why a pseudo-polynomial time algorithm does not necessarily run in polynomial time with respect to the input length. An NP-complete problem with known pseudo-polynomial time algorithms is called weakly NP-complete. An NP-complete problem is called strongly NP-complete if it is proven that it cannot be solved by a pseudo-polynomial time algorithm unless . The strong/weak kinds of NP-hardness are defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Operations Research
''Mathematics of Operations Research'' is a quarterly peer-reviewed scientific journal established in February 1976. It focuses on areas of mathematics relevant to the field of operations research such as continuous optimization, discrete optimization, game theory, machine learning, simulation methodology, and stochastic models. The journal is published by INFORMS (Institute for Operations Research and the Management Sciences). the journal has a 2017 impact factor of 1.078. History The journal was established in 1976. The founding editor-in-chief was Arthur F. Veinott Jr. (Stanford University). He served until 1980, when the position was taken over by Stephen M. Robinson, who held the position until 1986. Erhan Cinlar served from 1987 to 1992, and was followed by Jan Karel Lenstra (1993-1998). Next was Gérard Cornuéjols (1999-2003) and Nimrod Megiddo (2004-2009). Finally came Uri Rothblum (2009-2012), Jim Dai (2012-2018), and the current editor-in-chief Katya Scheinberg (20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" is decidable since there is a decision procedure called long division that gives the steps for determining whether ''x'' evenly divides ''y'' and the correct answer, ''YES'' or ''NO'', accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead. Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science. Applications Basic applications of combina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset Sum Problem
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' The problem is known to be NP-complete. Moreover, some restricted variants of it are NP-complete too, for example: * The variant in which all inputs are positive. * The variant in which inputs may be positive or negative, and T=0. For example, given the set \, the answer is ''yes'' because the subset \ sums to zero. * The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., T = \frac(a_1+\dots+a_n) . This special case of SSP is known as the partition problem. SSP can also be regarded as an optimization problem: find a subset whose sum is at most ''T'', and subject to that, as close as possible to ''T''. It is NP-hard, but there are several algorithms that can solve it reasonably q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Approximation Scheme
In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter and produces a solution that is within a factor of being optimal (or for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most , with being the length of the shortest tour. Sanjeev Arora, Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems, Journal of the ACM 45(5) 753–782, 1998. The running time of a PTAS is required to be polynomial in the problem size for every fixed ε, but can be different for different ε. Thus an algorithm running in time or even counts as a PTAS. Variants Deterministic A practical problem with PTAS algorithms is that the exponent of the polynomial co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
