 
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 MAXSAT 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 3/4approximation
algorithm.
(a) Show how to get a 3/4approximation by a deterministic algorithm.
(b) Combine the scheme and the random assignment in a slightly different way to get a randomized
0.77approximation algorithm for MAXSAT without clauses of size exactly 2.
3. Consider again the LP for the MAXSAT
(a) Show that if we randomly round each variable independently according to the function pi =
