15750
HW 3
1
15750 — Graduate Algorithms — Spring 2009
Miller and Dinitz and Tangwongsan
Assignment 3
Due date: Wednesday, March 18
Some Reminders:
•
Read the Policies section on the course web site before you start working on this assignment. Collab
oration
is
permitted for this assignment; however, you must write up your own solutions.
•
When solving these problems, you should refrain from looking up solutions from outside sources;
however, you should feel free to look up, say, Markov’s inequality or isoperimetric bounds. For each
problem, state whether you have seen it before. If you have questions, contact the course staff.
•
We
strongly
encourage you to type up your solutions (preferably using LaTeX). Extracredit will be
given if we use your wonderful writeup in our solutions. You may neatly handwrite your solutions,
but if we have trouble reading them, you will be required to type up future solutions.
•
When you give an algorithm, please also explain why it is correct and analyze its running time.
•
Start EARLY!
We encourage you to start working on this assignment early and take full advantage of
office hours.
1
Duality Theory
(25 pts.)
Given a linear program min
{
c
>
x
:
A
x
≥
b
,
x
≥
0
}
and a solution
x
0
, how can one decide whether
x
0
is an optimal solution? More generally, how can one calculate a
good
lower bound on such linear
programs? Duality is a key concept in linear programming that can help answer these questions.
Since we did not have time in class to cover such an important topic, you will learn about duality
theory firsthand in this problem.
More concretely, suppose we are given a linear program (which we will call the primal LP)
minimize
c
>
x
,
subject to
A
x
≥
b
and
x
≥
0
.
The following LP is called the
dual
linear program:
maximize
y
>
b
,
subject to
A
>
y
≤
c
and
y
≥
0
.
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, Dual problem, Duality, Strong Duality Theorem, Weak duality

Click to edit the document details