MAXPLANKellipsoid

MAXPLANKellipsoid - Two Lectures on the Ellipsoid Method...

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Unformatted text preview: Two Lectures on the Ellipsoid Method for Solving Linear Programs 1 Lecture 1: A polynomial-time algorithm for LP Consider the general linear programming problem: = max cx S.T. Ax b, x R n , (1) where A = ( a ij ) i,j Z m n is an m n integer matrix, and b = ( b i ) i Z m and c = ( c j ) j Z n are integer vectors. We denote by a T i the i th row of A . In the unit-cost model of computation, the cost of multiplying two integers x and y is cost( x y ) = 1. However, in the bit-model , it is cost( x y ) = ( x ) + ( y ), where ( x ) = log(1 + | x | ) + 1 = number of bits used to represent x in binary . For instance, consider the following code for computing x = a 2 k : 1. x a ; 2. for i = 1 ,...,k , set x x 2 . Then in the unit-cost model, the cost of this code is k , while in the bit-model, it is log x 2 k ( a ). Let = max { ( a ij ) , ( b i ) , ( c j ) } , and let L = summationdisplay i,j ( a ij ) + summationdisplay i ( b i ) + summationdisplay j ( c j ) be the total number of bits needed to represent the input. In the bit-model of computation, an algorithm is said to run in polynomial-time, if the number of operations ( { + , , ,/ } ) is at most poly( n,m, ) and the bit length of all numbers involved in the computation is at most poly( n,m, ). As we shall see, linear programming is polynomial-time solvable in the bit-model. * These notes were taken by Khaled Elbassioni (Max-Planck-Institut fur Informatik, Saarbrucken, Germany; (elbassio@mpi-sb.mpg.de)) while attending a graduate course on Lin- ear Programming taught by Leonid Khachiyan in Fall 1996 at Rutgers University. 1 The Ellipsoid method Let > 0 be a given constant. We call x an -approximate solution of (1) if c x and a T i x b i + , for i = 1 ,...,m . Assumption 1 We know R R + , such that (1) has an optimal solution x in the Euclidean ball B R = { x / bardbl x bardbl R } . Let h = max {| a ij | , | b i | , | c j |} , i.e., h = 2 1. Then under Assumption 1, the Ellipsoid method computes an -approximate solution of (1) in O (( n + m ) n 3 log ( Rhn ) ) arithmetic operations over O (log ( Rhn ) )-bit numbers. Fact 1 Let V be the unit ball V = { x / bardbl x bardbl 1 } , and let V = V { x / x n } . Then theres an ellipsoid E such that V E vol E vol V e 1 2( n +1) 1 1 2 n . Proof . Let the center of E be (0 , ,..., , 1 n +1 ) = 1 n +1 e n , and = 1 1 n +1 1 1 n (see Figure 1). The equation of E is parenleftBig x n 1 n +1 parenrightBig 2 2 + x 2 1 2 + x 2 2 2 + + x 2 n 1 2 1 ....
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MAXPLANKellipsoid - Two Lectures on the Ellipsoid Method...

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