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AMS/MBA 546 (Fall, 2010)
Estie Arkin
Network Flows
Homework Set # 7
Due Tuesday, November 9, 2010.
Suggested reading Chapters 6,7 of [AMO].
1). [6.47] In a directed acyclic network
G
, certain arcs are coloured blue. Consider the problem of covering
the blue arcs by directed paths, which can start and end at any node (these paths may contain blue arcs
and non blue arcs). Show that the minimum number of directed paths needed to cover the blue arcs is equal
to the maximum number of blue arcs that satisfy the property that no two of these arcs belong to the same
path. Will this result be valid if
G
contains directed cycles? (Hint: use the min ±ow max cut theorem.)
This is a theorem due to Dilworth. It is sometimes phrased using the following de²nitions: A partial order
≺
is a relation between elements
a
1
, a
2
, . . . , a
n
is such that it is transitive, asymetric (i.e., if
a
i
≺
a
j
then
a
j
n≺
a
i
), and for no
i a
i
≺
a
i
. If either
a
i
≺
a
j
or
a
j
≺
a
i
we say that the two elements are comparable.
As an example, you can think of the elements as being positive integers, and
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This note was uploaded on 01/31/2011 for the course AMS 546 taught by Professor Arkin,e during the Fall '08 term at SUNY Stony Brook.
 Fall '08
 Arkin,E

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