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Lecture12

# Lecture12 - C&O 355 Lecture 12 N Harvey Topics...

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C&O 355 Lecture 12 N. Harvey

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Topics Polynomial-Time Algorithms Ellipsoid Method Solves LPs in Polynomial Time Separation Oracles Convex Programs Minimum s-t Cut Example
Polynomial Time Algorithms P = class of problems that can be solved efficiently i.e., solved in time · n c , for some constant c , where n = input size This is a bit vague Consider an LP max { c T x : Ax · b } where A has size m x d Input is a binary file containing the matrix A, vectors b and c Two ways to define “input size” A. # of bits used to store the binary input file B. # of numbers in input file, i.e., m ¢ d + m + d Leads to two definitions of “efficient algorithms” A. Running time · n c where n = # bits in input file B. Running time · n c where n = m ¢ d + m + d “Polynomial Time Algorithm” Strongly Polynomial Time Algorithm”

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Algorithms for Solving LPs Unsolved Problems: Is there a strongly polynomial time algorithm? Does some implementation of simplex method run in polynomial time? Name Publication Running Time Practical? Fourier-Motzkin Elimination Fourier 1827, Motzkin 1936 Exponential No Simplex Method Dantzig '47 Exponential Yes Perceptron Method Agmon '54, Rosenblatt '62 Exponential Sort of Ellipsoid Method Khachiyan '79 Polynomial No Interior Point Method Karmarkar '84 Polynomial Yes Analytic Center Cutting Plane Method Vaidya '89 & '96 Polynomial No Random Walk Method Bertsimas & Vempala '02-'04 Polynomial Probably not Boosted Perceptron Method Dunagan & Vempala '04 Polynomial Probably not Random Shadow-Vertex Method Kelner & Spielman '06 Polynomial Probably not
Recall how the algorithm works: It starts at a vertex of the polyhedron It moves to a “neighboring vertex” with better objective value It stops when it reaches the optimum How many moves can this take? For any polyhedron, and for any two vertices, can you move between them with few moves ? Why is analyzing the simplex method hard?

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For any polyhedron, and for any two vertices, can you move between them with few moves ? The Hirsch Conjecture (1957) Let P = { x : Ax · b } where A has size m x n. You can move between any two vertices using only m-n moves. Example: A cube. Dimension n=3. # constraints m=6. Do m-n=3 moves suffice? Yes! Why is analyzing the simplex method hard?
Why is analyzing the simplex method hard? For any polyhedron, and for any two vertices, can you move between them with few moves ? The Hirsch Conjecture (1957) Let P = { x : Ax · b } where A has size m x n. You can move between any two vertices using only m-n moves. We have no idea how to prove this. Theorem: [Kalai-Kleitman 1992] m log n+2 moves suffice. Still the best known result. Proof amazingly beautiful ! We might prove it later in the course… Want to prove a better bound? A group of (eminent) mathematicians have a blog organizing a massively collaborative project to do just that.

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Ellipsoid Method for Solving LPs Ellipsoid method finds feasible point in P = { x : Ax · b } i.e., it can solve a system of inequalities
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Lecture12 - C&O 355 Lecture 12 N Harvey Topics...

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