Lecture8 - C&O 355 Mathematical Programming...

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Unformatted text preview: C&O 355 Mathematical Programming Fall 2010 Lecture 8 N. Harvey Polynomial Time Algorithms P = { computational 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 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 The Genius behind the Ellipsoid Method Intelligence gathered by this and other governments leaves no doubt that the Iraq regime continues to possess and conceal some of the most lethal weapons ever devised George W. Bush, 3/18/2003 WMD in Iraq We are learning more as we interrogate or have discussions with Iraqi scientists and people within the Iraqi structure, that perhaps he destroyed some, perhaps he dispersed some. And so we will find them. George W. Bush, 4/24/2003 Finding WMD USA have a satellite with a WMD detector The detector scans a round region of the earth It can compare two halves of the region, and decide which half is more likely to have WMD Finding WMD USA have a satellite with a WMD detector The detector scans a round region of the earth It can compare two halves of the region, and decide which half is more likely to have WMD It continues by rescanning the more likely half Finding WMD USA have a satellite with a WMD detector The detector scans a round region of the earth...
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Lecture8 - C&O 355 Mathematical Programming...

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