 
Summary: Industrial Engineering & Operations Research, UC Berkeley
IEOR269 Integer Programming and Combinatorial Optimization
Semester: Spring 2004 Instructor: Alper Atamt¨urk
Midterm Exam Due: March 17, 2004, 5PM (4175 Etcheverry)
1. Let A be a rational valued m × n matrix and b be a rational valued mdimensional
column vector. Let P = {x IRn
: Ax b} be a nonempty polyhedron and x P.
(a) Prove that for x, the violation of any Chv´atalGomory (CG) inequality =
yT Ax  yT b , where y IRm
+ , yT A integral, is upper bounded by a constant.
(b) Let s = b  Ax and call a CG inequality complementary (with respect to x)
if yis
i = 0 for all i = 1, . . . , m. Prove that there is a polynomial algorithm that
determines whether there is a complementary CG inequality violated by x or
not (and in the former case outputs such an inequality).
2. Consider the affine transformation T(x) = QA1/2(x  a) defined in the ellipsoid
method of Khachiyan. Prove that
T( ^E(A, a)) = ^E(I, 0).
3. Prove that if H and G are two faces of a polyhedron P of dimension r and r + s
(r 0, s > 0 and both integer), respectively, and H is a face of G, then there exists
