Exercises
145
In the following program the variable
x
ij
will be selected to be 1 if
man
i
and woman
j
are matched in the matching selected:
maximize
i,j
x
ij
subject to
j
x
m,j
≤
1 for all men
m
(E7.1)
i
x
i,w
≤
1 for all women
w
j>
m
w
x
m,j
+
i>
w
m
x
i,w
+
x
m,w
≥
1 for all pairs (
m, w
)
x
m,w
∈
{
0
,
1
}
for all pairs (
m, w
)
•
Prove that this integer program is a correct formulation of the
stable matching problem.
•
Consider the relaxation of the integer program that allows
frac-
tional
stable matchings. It is identical to the above program, ex-
cept that instead of each
x
m,w
being either 0 or 1,
x
m,w
is allowed
to take any real value in [0
,
1]. Show that the following program
is the dual program to the relaxation of E7.1.
minimize
i
α
i
+
j
β
j
−
i,j
γ
ij
subject to
α
m
+
β
w
−
j<
m
w
γ
m,j
−
i<
w
m
γ
i,w
−
γ
m,w
≥
1
for all pairs (
m, w
)
α
i
,
β
j
,
γ
i,j
≥
0 for all
i
and
j
.
•
Use complementary slackness (Theorem
??
) to show that every
feasible fractional solution to the relaxation of E7.1 is optimal
and that setting
α
m
=
j
x
m,j
for all
m
,
β
w
=
i
x
i,w
for all
w
and
γ
ij
=
x
ij
for all
i, j
is optimal forthe dual program.