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Jack R. Edmonds
Jack R. Edmonds (born April 5, 1934) is an American-born and educated computer scientist and mathematician who lived and worked in Canada for much of his life. He has made fundamental contributions to the fields of combinatorial optimization, polyhedral combinatorics, discrete mathematics and the theory of computing. He was the recipient of the 1985 John von Neumann Theory Prize. Early career Edmonds attended Duke University before completing his undergraduate degree at George Washington University in 1957. He thereafter received a master's degree in 1960 at the University of Maryland under Bruce L. Reinhart with a thesis on the problem of embedding graphs into surfaces. From 1959 to 1969 he worked at the National Institute of Standards and Technology (then the National Bureau of Standards), and was a founding member of Alan Goldman’s newly created Operations Research Section in 1961. Goldman proved to be a crucial influence by enabling Edmonds to work in a RAND Corporation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polymatroid
In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. It is also described as the multiset analogue of the matroid. Definition Let E be a finite set and f: 2^E\rightarrow \mathbb_+ a non-decreasing submodular function, that is, for each A\subseteq B \subseteq E we have f(A)\leq f(B) , and for each A, B \subseteq E we have f(A)+f(B) \geq f(A\cup B) + f(A\cap B) . We define the polymatroid associated to f to be the following polytope: P_f= \Big\. When we allow the entries of \textbf to be negative we denote this polytope by EP_f, and call it the extended polymatroid associated to f. An equivalent definition Let E be a finite set and \textbf, \textbf \in \mathbb^E. We call the ''modulus'' of \textbf to be the sum of all of its entries, denoted , \textbf, , and denote \textbf \leq \textbf whenever v_i-u_i\geq 0 for every i \in E (notice that this gives an order to \mathbb_+^E). A polymat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Matching
Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph , and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible. An important special case of the maximum cardinality matching problem is when is a bipartite graph, whose vertices are partitioned between left vertices in and right vertices in , and edges in always connect a left vertex to a right vertex. In this case, the problem can be efficiently solved with simpler algorithms than in the general case. Algorithms for bipartite graphs Flow-based algorithm The simplest way to compute a maximum cardinality matching is to follow the Ford–Fulkerson algorithm. This algorithm solves the more general problem o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blossom Algorithm
In graph theory, the blossom algorithm is an algorithm for constructing maximum matchings on graphs. The algorithm was developed by Jack Edmonds in 1961, and published in 1965. Given a general graph , the algorithm finds a matching such that each vertex in is incident with at most one edge in and is maximized. The matching is constructed by iteratively improving an initial empty matching along augmenting paths in the graph. Unlike bipartite matching, the key new idea is that an odd-length cycle in the graph (blossom) is contracted to a single vertex, with the search continuing iteratively in the contracted graph. The algorithm runs in time , where is the number of edges of the graph and is its number of vertices. A better running time of O( , E, \sqrt ) for the same task can be achieved with the much more complex algorithm of Micali and Vazirani. A major reason that the blossom algorithm is important is that it gave the first proof that a maximum-size matching could be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RAND Corporation
The RAND Corporation (from the phrase "research and development") is an American nonprofit global policy think tank created in 1948 by Douglas Aircraft Company to offer research and analysis to the United States Armed Forces. It is financed by the Federal government of the United States, U.S. government and private Financial endowment, endowment, corporations, university, universities and private individuals. The company assists other governments, international organizations, private companies and foundations with a host of defense and non-defense issues, including healthcare. RAND aims for interdisciplinary and quantitative problem solving by translating theory, theoretical concepts from formal economics and the Outline of physical science, physical sciences into novel applications in other areas, using applied science and operations research. Overview RAND has approximately 1,850 employees. Its American locations include: Santa Monica, California (headquarters); Arlington ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan J
Alan may refer to: People *Alan (surname), an English and Turkish surname *Alan (given name), an English given name **List of people with given name Alan ''Following are people commonly referred to solely by "Alan" or by a homonymous name.'' *Alan (Chinese singer) (born 1987), female Chinese singer of Tibetan ethnicity, active in both China and Japan *Alan (Mexican singer) (born 1973), Mexican singer and actor *Alan (wrestler) (born 1975), a.k.a. Gato Eveready, who wrestles in Asistencia Asesoría y Administración *Alan (footballer, born 1979) (Alan Osório da Costa Silva), Brazilian footballer *Alan (footballer, born 1998) (Alan Cardoso de Andrade), Brazilian footballer *Alan I, King of Brittany (died 907), "the Great" *Alan II, Duke of Brittany (c. 900–952) *Alan III, Duke of Brittany(997–1040) *Alan IV, Duke of Brittany (c. 1063–1119), a.k.a. Alan Fergant ("the Younger" in Breton language) *Alan of Tewkesbury, 12th century abbott *Alan of Lynn (c. 1348–1423), 15th cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics". The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Combinatorics
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex). Additionally, many computer scientists use the phrase “polyhedral combinatorics” to describe research into precise descriptions of the faces of certain specific polytopes (especially 0-1 polytopes, whose vertices are subsets of a hypercube) arising from integer progr ... [...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. Some research literature considers discrete o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Scientist
A computer scientist is a person who is trained in the academic study of computer science. Computer scientists typically work on the theoretical side of computation, as opposed to the hardware side on which computer engineers mainly focus (although there is overlap). Although computer scientists can also focus their work and research on specific areas (such as algorithm and data structure development and design, software engineering, information theory, database theory, computational complexity theory, numerical analysis, programming language theory, computer graphics, and computer vision), their foundation is the theoretical study of computing from which these other fields derive. A primary goal of computer scientists is to develop or validate models, often mathematical, to describe the properties of computational systems (processors, programs, computers interacting with people, computers interacting with other computers, etc.) with an overall objective of discovering des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
