PRELIM 3
ORIE 321/521
April 15, 2008
Closed book exam. Justify all work.
1.
(a) (20 points) Suppose the members of
n
committees
C
1
, . . . , C
n
are drawn from a common pool
of individuals
i
= 1
, . . . , m
; i.e.,
C
j
⊆ {
1
, . . ., m
}
, for
j
= 1
, . . . , n
, where
C
j
denotes the set of
individuals who are on the
j
th committee. From the membership of each committee we would
like to select a person to serve as committee chair. Moreover, in order to distribute the workload
evenly, we wish to know whether it is possible to select
distinct
chairs for the committees, i.e.,
n
different
individuals
i
1
, . . . , i
n
with
i
j
∈
C
j
for
j
= 1
, . . . , n
. Indicate a
bipartite cardinality
matching model
which can be used to determine whether such a selection of individuals to chair
the
n
committees is possible. You must interpret the vertices and edges of your bipartite graph
and explain clearly why the solution for your model actually determines the individuals to chair
the various committees or proves that a complete selection of
n
distinct chairs is impossible.
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 Spring '10
 TROTTER
 Ethnic group, Flow network, Maximum flow problem, j th committee, distinct chairs

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