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CS 573
Final Exam Questions
Fall 2010
This exam lasts 180 minutes.
Write your answers in the separate answer booklet.
Please return this question sheet with your answers.
1. A subset
S
of vertices in an undirected graph
G
is called
trianglefree
if, for every triple of vertices
u
,
v
,
w
∈
S
, at least one of the three edges
uv
,
uw
,
vw
is
absent
from
G
.
Prove
that ﬁnding the size
of the largest trianglefree subset of vertices in a given undirected graph is NPhard.
A trianglefree subset of 7 vertices.
This is
not
the largest trianglefree subset in this graph.
2. An
n
×
n
grid is an undirected graph with
n
2
vertices organized into
n
rows and
n
columns. We
denote the vertex in the
i
th row and the
j
th column by
(
i
,
j
)
. Every vertex in the grid have exactly
four neighbors, except for the
boundary
vertices, which are the vertices
(
i
,
j
)
such that
i
=
1,
i
=
n
,
j
=
1, or
j
=
n
.
Let
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
,...,
(
x
m
,
y
m
)
be distinct vertices, called
terminals
, in the
n
×
n
grid. The
escape problem
is to determine whether there are
m
vertexdisjoint paths in the grid that connect
the terminals to any
m
distinct boundary vertices. Describe and analyze an efﬁcient algorithm to
solve the escape problem.
A positive instance of the escape problem, and its solution.
3. Consider the following problem, called U
NIQUE
S
ET
C
OVER
. The input is an
n
element set
S
, together
with a collection of
m
subsets
S
1
,
S
2
,...,
S
m
⊆
S
, such that each element of
S
lies in exactly
k
subsets
S
i
. Our goal is to select some of the subsets so as to maximize the number of elements of
S
that lie in
exactly one
selected subset.
(a) Fix a real number
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This note was uploaded on 01/22/2012 for the course CS 573 taught by Professor Chekuri,c during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Chekuri,C

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