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CME 305: Discrete Mathematics and Algorithms
Instructor: Professor Amin Saberi (saberi@stanford.edu)
HW#3 – Due 03/02/10
1. Recall the deﬁnition of totally unimodularity: a matrix
A
is totally unimodular if and
only if the determinant of every submatrix of
A
is in
{
1
,
0
,
1
}
.
(a) Show that if every column of
A
has at most one +1 and at most one

1, and all
other entries are zero, then
A
is totally unimodular.
(b) Let
G
(
V,E
) be a directed graph, and assume there is no edge entering
s
, and no
edge leaving
t
. Write down a linear program for computing the maximum
s

t
ﬂow
in
G
and show that the constraint matrix of the LP is totally unimodular.
2. Recall the maximum cut problem. Given a graph
G
(
V,E
) we want to partition the
graph into two (disjoint) sets
A
and
B
such that the number of edges between them
(the edges with one end point in
A
and the other in
B
) is maximized.
Consider the following algorithm. We start with an arbitrary partition (
A,B
). If there
exists a vertex
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 Winter '09

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