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# 04 - Spring 2010 CS530 Analysis of Algorithms Homework 4...

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Spring 2010 CS530 – Analysis of Algorithms Homework 4 Homework 4, due Feb 24 You must prove your answer to every question. Problems with a ( * ) in place of a score may be a little too advanced, or too challenging to most students, so I do not assign a score to them. But I will still note if you solve them. Problem 1. (10pts) Let Ax = b be a solvable set of m linear equations with n variables, where each entry of A , b , is an integer with at most k bits. Show that there is a solution in which each coordinate can be bounded by 2 L where L is bounded by a polynomial in k , m . [ Hint: recall Cramer’s Rule, or our analysis of the exact computation of Gaussian elimination. ] Solution. Let the rank be r . Choose an independent subset of r equations. We know that n - r variables can be chosen as free parameters. Fix the value of these variables to, say, 0. Now we have an r × r system of equations that can be solved by Cramer’s Rule. In the solution each x i is the quotient of two determinants chosen from the original coefficients. The numerator determinant | d i j | can be estimated, using the Hadamard inequality, as i ( j d 2 i j ) 1 / 2 . Since each d i j is bounded by 2 k , the total is bounded by ( r · 2 2 k ) r / 2 = r r / 2 2 kr = 2 r ( k + 1 2 log r ) . So we can choose L = r ( k + 1 2 log r ) < m ( k + m ) . Problem 2. (10pts) Let the constraints of a linear program be given by Ax b , and the objective function to maximize by c T x . Suppose that the problem is feasible. Show that if there is no optimal solution then we can add some more constraints of the type x j d j or x j d j (at most one for each j ) in such a way that the new system has an optimal solution.

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04 - Spring 2010 CS530 Analysis of Algorithms Homework 4...

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