Primal Dual

Primal-Dual Approximation Algorithms

We just saw how the primal-dual schema permits sometimes designing efficient combinatorial algorithms for solving certain problems. We will now see an example of how a related technique can sometimes be used to design efficient approximation algorithms
The major tool that we will use will be the
RELAXED Complementary Slackness conditions
The problem we examine will again be weighted set-cover.

1

Recall that given canonical primal

n

minimize

cj xj

subject to

j =1 a′ x ≥ bi, i xj ≥ 0,

i = 1, . . . , m j = 1, . . . , n

the dual is m maximize subject to

biπi i=1 πAj ≤ cj ,

πi ≥ 0,

j = 1, . . . , n i = 1, . . . , m

2

Theorem (Complementary Slackness):
Let x and π respectively be primal and dual feasible solutions. Then x and π are both optimal if and only if all of the following conditions are satisfied.

Primal Complementary Slackness conditions
∀1 ≤ j ≤ n : either xj = 0 or π ′Aj = cj
Dual Complementary Slackness conditions
∀1 ≤ i ≤ m : either πi = 0 or a′ x = bi i 3

Theorem (RELAXED Complementary Slackness):
Let x and y respectively be primal and dual feasible solutions. Suppose further that for some α > 1, x and y satisfy all of
Primal Complementary Slackness conditions
∀1 ≤ j ≤ n : either xj = 0 or πAj = cj
RELAXED Dual C.S. conditions
∀1 ≤ i ≤ m : either πi = 0 or a′ x ≤ αbi i Then m n

cj xj ≤ α · j =1

biπi i=1 Proof:

j =1

a′ x i (πAj ) xj = πAx =

cj xj = j =1

m

m

n

n

i=1

πi ≤ α

biπi. i=1 Given such an x, π we immediately know that x is within α of OPT, the minimum cost optimum solution.
4

Recall Weighted Set Cover problem where each set
F has a weight Cost(F ) = C (F ), and the problem is to find a Set Cover of C of Minimum Weight,
Cost(C ) = F ∈C C (F ).
For example X = {1, 2, 3, 4, 5, 6} and F contains the subsets
F1
F2
F3
F4
F5

=
=
=
=
=

{1, 3, 5};
{2, 3,