Industrial Engineering
Fall
Integer Programming
Homework
Last updated: October
,
Due: Friday October
, in class
Note: only a proper subset of these problems will be graded.
Prob em .
(Bertsimas and Weismantel, Exercise
. ) Prove that a matrix of zeroone elements, in which each
column has consecutive ones, is totally unimodular.
Prob em
.
(Bertsimas and Weismantel, Exercise
. ) Let
b
∈
Z
m
. Show that
{
x
∈
R
n
∶
Ax
≤
b
,
x
≥
}
is
integral if and only if
{(
x
,
s
)
∈
R
n
+
m
∶
Ax
+
s
=
b
,
x
,
s
,
≥
}
is integral.
Prob em
.
(Bertsimas and Weismantel, Exercise
. )
a. Show that an incidence matrix of a general undirected graph is not totally unimodular.
b.
Show that an incidence matrix of an undirected graph is totally unimodular if and only if the graph is
bipartite.
(See Chapter
of Korte and Vygen for a formal de nition of the incidence matrix of an undirected graph.)
Prob em
.
(Bertsimas and Weismantel, Exercise
.
)
a.
Recall the integer programming formulation of the minimum cut problem on an undirected graph we
discussed in Lecture
. Show how to modify this formulation to
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 Spring '97
 RAVINDRAN
 Linear Programming, Industrial Engineering, Graph Theory, Optimization, undirected graph, Bipartite graph

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