IE 418 – Integer Programming
Problem Set #3
Due Date: April 4
Do the following problems. You may work with one other classmate. If you work alone,
you will receive a 10% bonus. You are allowed to examine outside sources, but you must cite
any references that you use.
Please don’t discuss the problems with other members
of the class (other than your partner, if you are working with one).
1
Reductions
These Problems are
NP
Complete
PROBLEM:
Number Partition Problem (NPP)
INSTANCE:
Finite set
N
of items, each with size
s
n
∈
Z
+
∀
n
∈
N
, and an
integer
K
.
QUESTION:
Does there exist a subset
N
⊆
N
such that
∑
n
∈
N
s
n
=
∑
n
∈
N
\
N
s
n
?
PROBLEM:
Vertex Cover Problem (VCP)
INSTANCE:
Graph
G
= (
V, E
), and a positive integer
K
≤ 
V

.
QUESTION:
Is there a vertex cover of size K or less in G. (Does there exist
a subset
V
⊆
V
with

V
 ≤
K
such that for each edge
e
=
(
u, v
)
∈
E
either
u
∈
V
or
v
∈
V
)?
1.1 Problem
Show that the Dominating Set Problem (DSP) is
NP
Complete:
PROBLEM:
Dominating Set Problem (DSP)
INSTANCE:
Graph
G
= (
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 Spring '08
 Ralphs
 Natural number, NPcomplete problems, independent set, Prof Jeff Linderoth

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