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Solution Set for CS 482, Prelim 2
April 8, 2008
Questions in red, solutions in black.
PROBLEM 1
(20 points)
PART A
(15 points)
Find a maximum flow and minimum st cut in
the flow network G shown here.
The source
and sink are s and t, respectively.
The edge
capacities are denoted by numbers
alongside the edges in the diagram.
You may express the minimum st cut by
writing a partition of the vertex set
{s,v,w,x,y,t} into two sets (A,B).
A =
{s,w,x,y}
B =
{v,t}
The flow shown at right is one of the three
integer maximum flows in this network.
Any
flow of value 21 constitutes a correct
solution.
s
w
y
t
v
x
10
8
6
2
1
3
2
10
8
9
s
w
y
t
v
x
9
9
0
8
8
10
2
2
4
0
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View Full Document PART B
(5 points)
Give a counterexample to the following statement.
Let G be a flow network with source s and sink t. Suppose f is a
maximum flow in G and (A,B) is a minimum st cut in G. If the
capacity of every edge of G is increased by 1, then the maximum
flow value increases from v(f) to v(f)+k, where k is the number
of edges from A to B, i.e. the number of edges (u,v) with u in A
and v in B.
Your counterexample should consist of an explicit flow network G that
violates the statement. You should describe G (the vertices, edges, source
and sink nodes, and edge capacities) in words, in pictures, or both. If you
describe G using a diagram, make sure that all vertices and edges of G can
be clearly identified in the diagram.
There are many counterexamples. Here is one. In this flow network the
maximum flow value is v(f)=3, and the unique minimum st cut is (A,B),
where A={s} and B={w,x,y,z,t}. Hence k=3, and v(f)+k=6. If we increase
the capacity of each edge by 1, we obtain a new flow network (shown in the
second diagram) whose maximum flow value is 5, not 6.
y
t
4
2
2
1
w
1
z
1
x
s
y
t
5
3
3
2
w
2
3
z
2
x
s
2
PROBLEM 2
(20 points)
FordFulkerson University has many universitywide committees that need to
be filled with professors.
The university has n professors and m committees.
‣
Each committee k has a required number of members, r
k
.
‣
No professor is allowed to serve on more than c committees.
‣
For each professor j, there is a list L
j
of committees on which he or she
is qualified to serve.
(a)
[12 points] Design a polynomialtime algorithm to determine whether it
is possible to fill each committee with the required number of qualified
professors.
(b)
[8 points] FordFulkerson University is organized into d departments,
and each professor belongs to only one department. The university
institutes a new rule that no committee is allowed to have more than one
professor from the same department. Design a polynomialtime
algorithm to determine whether it is possible to fill each committee with
the required number of qualified professors from different departments.
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This note was uploaded on 10/02/2008 for the course CS 482 taught by Professor Kleinberg during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 KLEINBERG
 Algorithms

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