Algorithmic Methods Fall Semester, 2010/11 Exercise 2: November 29, 2010 Summary: Algorithmic Methods Fall Semester, 2010/11 Exercise 2: November 29, 2010 Lecturer: Prof. Yossi Azar Write short but full and accurate answers. Each question should start on a new separate page and each of its parts should not exceed a page. 1. (a) An n by m matrix A is called Monge matrix if for all 1 i n - 1 and 1 j m - 1 we have Ai+1,j+1 + Ai,j Ai,j+1 + Ai+1,j. The distance between two matrices A and B is defined to be i,j |Ai,j - Bi,j|. Given a matrix B find a Monge matrix A closest (i.e minimum distance) to A. (b) An n by n matrix A is called Balanced if the sum of the entries of any n/2 rows (columns, respectively) is at most twice the sum of the entries of any n/2 columns (columns, respectively). Given a matrix B find a Balanced matrix A closest (i.e minimum distance) to A. 2. Consider the 1 - 1/e approximation algorithm for MAX-SAT which is based on LP and randomly rounding each variable independently according to the function pi = xi where x is the LP optimal fractional solution. The scheme was combined with random assignment to get a 3/4-approximation algorithm. (a) Show how to get a 3/4-approximation by a deterministic algorithm. (b) Combine the scheme and the random assignment in a slightly different way to get a randomized 0.77-approximation algorithm for MAX-SAT without clauses of size exactly 2. 3. Consider again the LP for the MAX-SAT (a) Show that if we randomly round each variable independently according to the function pi = Collections: Computer Technologies and Information Sciences