match-up between any two players.
Now every pair of players can go into a match-up, and the
benefit incurred by this match-up will be indicated by the metric
B
(
Match
). Coach P would like
to set up one-on-one games between the players such that the total amount of benefits from all
these match-ups is maximized. Can you help him do that through an ILP?

3) This is a modification of the problem in 2). The first modification is that the resulting graph
connecting players is a complete graph. In addition, each edge
e
∈ E
has an associated weight
w
e
equal to the corresponding value of
B
(
Match
) for the match-up between the corresponding players.
Then the problem can be expressed as the following ILP
max
b
∑
e
∈E
w
e
b
e
subject to
∑
e
∈
N
(
v
)
b
e
≤
1
,
∀
v
∈ {
1
,
· · ·
,
|V|}
b
e
∈ {
0
,
1
}
.

Problem 2
(
4 points
): Consider a graph
G
= (
V
,
E
). A
matching
on
G
is a collection of
M ⊆ E
such
that, no two edges in
M
share a vertex. In other words, each vertex in
V
has at most one connected
edge in
M
. A
maximal matching
is a matching
M
such that, if any other edge in
E \ M
is added
to
M
it no longer becomes a valid matching.
Assume that you have an algorithm that takes as input an arbitrary graph
G
, and outputs a maximal
matching
M
. Propose a heuristic that takes as input
G
and
M
, and outputs a valid vertex cover
C
(a cover is a subset of vertices
C ⊆ V
such that each edge in
E
is incident to at least one vertex
in
C
). The size of the output vertex cover should be within an approximation factor of 2.