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.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Linear Programming, Algorithms, Optimization, Elementary algebra, Analysis of algorithms, Ax B

Click to edit the document details