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
(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 =