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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
The knapsack problem is the following problem in combinatorial optimization:
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome in a mathematical model whose requirements and objective are represented by linear relationships. Linear programming is a special case of mathematical programming.
In computer science, the clique problem is the computational problem of finding cliques in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique, finding a maximum weight clique in a weighted graph, listing all maximal cliques, and solving the decision problem of testing whether a graph contains a clique larger than a given size.
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, formal language theory, the lambda calculus and type theory.
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output are random variables.
The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. Originally, the prizes were paid out of a memorial fund administered by the AMS that was established by friends of the late Delbert Ray Fulkerson to encourage mathematical excellence in the fields of research exemplified by his work. The prizes are now funded by an endowment administered by MPS.
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically:
The Birkhoff polytopeBn is the convex polytope in RN whose points are the doubly stochastic matrices, i.e., the n × n matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1. It is named after Garrett Birkhoff.
In mathematics, a unique sink orientation is an orientation of the edges of a polytope such that, in every face of the polytope, there is exactly one vertex for which all adjoining edges are oriented inward. If a polytope is given together with a linear objective function, and edges are oriented from vertices with smaller objective function values to vertices with larger objective values, the result is a unique sink orientation. Thus, unique sink orientations can be used to model linear programs as well as certain nonlinear programs such as the smallest circle problem.
In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the vertices of a convex polytope specified by a set of linear inequalities:
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.
Nimrod Megiddo is a mathematician and computer scientist. He is a research scientist at the IBM Almaden Research Center and Stanford University. His interests include combinatorial optimization, algorithm design and analysis, game theory, and machine learning. He was one of the first people to propose a solution to the bounding sphere and smallest-circle problem.
In the study of algorithms, an LP-type problem is an optimization problem that shares certain properties with low-dimensional linear programs and that may be solved by similar algorithms. LP-type problems include many important optimization problems that are not themselves linear programs, such as the problem of finding the smallest circle containing a given set of planar points. They may be solved by a combination of randomized algorithms in an amount of time that is linear in the number of elements defining the problem, and subexponential in the dimension of the problem.
In mathematics, economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions to an instance of the stable matching problem.
In convex geometry and polyhedral combinatorics, the extension complexity of a convex polytope is the smallest number of facets among convex polytopes that have as a projection. In this context, is called an extended formulation of ; it may have much higher dimension than .
Werner Römisch is a German mathematician, professor emeritus at the Humboldt University of Berlin, most known for his pioneer work in the field of stochastic programming.
Reverse-search algorithms are a class of algorithms for generating all objects of a given size, from certain classes of combinatorial objects. In many cases, these methods allow the objects to be generated in polynomial time per object, using only enough memory to store a constant number of objects. They work by organizing the objects to be generated into a spanning tree of their state space, and then performing a depth-first search of this tree.
References
- 1 2 3 4 5 "Eitan Zemel's profile at NYU Stern School of Business". Archived from the original on 2010-06-13. Retrieved 2009-02-18.
- ↑ "Master of Science in Business Analytics".
- ↑ Eitan Zemel's online publications resume
