CME305 Sample Midterm II
1.
Matchings and Vertex Covers
(a) Define what a matching in
G
is.
(b) Define what a vertex cover of
G
is.
(c) Let
M
be a maximum matching and C a minimum vertex cover.
Show that

M
 ≤ 
C
 ≤
2

M

.
Solution:
1. A matching
M
is a subset of
E
such that no two edges share an end
point.
2. A vertex cover
C
is a subset of
V
such that all edges are incident to at
least one element of
C
.
3. Let
M
be a maximum matching and
C
a minimum vertex cover. It is
easy to see that we need at least

M

nodes to cover all the edges of
M
. Hence

M
 ≤ 
C

. Now let us prove that
V
(
M
), the vertices in the
matching
M
, is a vertex cover. Assume it isn’t, then there is at least one
edge
e
that is not covered by
V
(
M
). It is easy to see that this implies
that
M
∪ {
e
}
is a matching so
M
is not maximum.
Contradiction.
Furthermore, we know that

V
(
M
)

= 2

M

, hence

C
 ≤
2

M

.
2.
Traveling Salesman Problem
Assume that deciding whether a graph has a Hamiltonian cycle is NP
Complete. Prove that the Traveling Salesman Problem is NPHard.
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 Winter '09
 vertex cover, NPcomplete problems, Computational problems in graph theory, Travelling salesman problem, NPcomplete, TSP

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