lecture14 - CS787 Advanced Algorithms Scribe Amanda Burton...

CS787: Advanced Algorithms
Scribe: Amanda Burton, Leah Kluegel Lecturer: Shuchi Chawla
Topic: Primal-Dual Algorithms Date: 10-17-07
14.1 Last Time
We finished our discussion of randomized rounding and began talking about LP Duality.
14.2 Constructing a Dual
Suppose we have the following primal LP.
min
∑
i
c
i
x
i
s.t.
∑
i
A
ij
x
i
≥ b
j
∀ j = 1 , ..., m
x
i
≥ 0
In this LP we are trying to minimize the cost function subject to some constraints. In considering
this canonical LP for a minimization problem, lets look at the following related LP.
max
∑
j
b
j
y
j
s.t.
∑
j
A
ij
y
j
≤ c
i
∀ i = 1 , ..., n
y
j
≥ 0
Here we have a variable y
j
for every constraint in the primal LP. The objective function is a linear
combination of the b
j
multiplied by the y
j
. To get the constraints of the new LP, if we multiply
each of the constraints of the primal LP by the multiplier y
j
, then the coefficients of every x
i
must
sum up to no more than c
i
. In this way we can construct a dual LP from a primal LP.
14.3 LP Duality Theorems
Duality gives us two important theorems to use in solving LPs.
Theorem 14.3.1 (Weak LP Duality Theorem) If x is any feasible solution to the primal and
y is any feasible solution to the dual, then V al
P
( x ) ≥ V al
D
( y ) .
1

Figure 14.3.1: Primal vs. Dual in the Weak Duality Theorem.
On the number line we see that all possible values for dual feasible solutions lie to the left of all
possible values for primal feasible solutions.
Last time we saw an example in which the optimal value for the dual was exactly equal to some
value for the primal. This introduces the question: was it a coincidence that this was the case?
The following theorem claims that no, it was not a coincidence. In fact, this is always the case.
Theorem 14.3.2 (Strong LP Duality Theorem) When P and D have non-empty feasible re-
gions, then their optimal values are equal, i.e. V al
*
P
= V al
*
D
.
On the number line we see that the maximum value for dual feasible solutions is equivalent to the
minimum value for primal feasible solutions.
Figure 14.3.2: Primal vs. Dual in the Strong Duality Theorem.
It could happen that there is some LP with no solution that satisfies all of the constraints. In this
case the feasible region would be empty. In this case the LP’s dual will be unbounded in the sense
that you can achieve any possible value in the dual.
The Strong LP Duality Theorem has a fairly simple geometric proof that will not be shown here
due to time constraints. Recall from last time the proof of the Weak LP Duality Theorem.
2

Proof: (Theorem 14.3.1)
Start with some feasible solution to the dual LP, say y . Let y ’s objective function value be V al
D
( y ).
Let x be a feasible solution to the primal LP with objective function value V al
P
( x ). Since y is a
feasible solution for the dual and x is a feasible solution to the primal we have
V al
D
( y ) =
∑
j
b
j
y
j
≤
∑
j
(
∑
i
A
ij
x
i
) y
j
=
∑
i
∑
j
A
ij
x
i
y
j
=
∑
i
(
∑
j
A
ij
y
j
) x
i
≤
∑
i
c
i
x
i
= V al
P
( x )
So just rewriting the equations shows us that the value of the dual is no more than the value of the
primal. What happens if these values are exactly equal to one another? This happens when