Primal Dual Method

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Primal Dual Method
found x, succeed!
primal
dual
restricted primal
restricted dual
y
z
Construct a better dual

• Lecture 20 March 28

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Primal Dual Program
Primal Program
Dual Program
If there is a feasible primal solution x and a
feasible dual solution y, then both are optimal
solutions.
Primal-Dual Method An algorithm to construct
such a pair of solutions.
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Optimality Condition
Suppose there is a feasible primal solution x,
and a feasible dual solution y. How do we check
that they are optimal solutions?
Avoid strict inequality
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Optimality Condition
Avoid strict inequality
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Complementary Slackness Conditions
Primal complementary slackness condition
Dual complementary slackness condition
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Primal Dual Method
Start from a feasible dual solution. Search for
a feasible primal solution satisfying
complementary slackness conditions. If not,
improve the objective value of dual solution.
repeat
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Restricted Primal
Given a feasible dual solution y, how do we
search for a feasible primal solution x that
satisfies complementary slackness conditions?
Formulate this as an LP itself!
Primal complementary slackness condition
Dual complementary slackness condition
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Restricted Primal
If j not in J, then we need x(j) to be zero. If
i not in I, then we need
If zero, we are done
Restricted primal
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Restricted Dual
Suppose the objective of the restricted primal is
not zero, what do we do?
Then we want to find a better dual solution.
nonzero
Restricted primal
Restricted dual
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Restricted Dual
Consider y?z as the new dual solution.
nonzero
Restricted dual
Dual Program
Larger value
Still feasible
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General Framework
found x, succeed!
primal
dual
restricted primal
restricted dual
y
z
Construct a better dual
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Bipartite Matching
Primal complementary slackness condition
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Hungarian Method
Start from a feasible vertex cover
Consider y- ?z, a better dual
Find a perfect matching using tight edges
non-zero
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Remarks
It is not a polynomial time method. It reduces
the weighted problem to the unweighted
problem, so that the restricted primal linear
program is easier to solve, and often there are
combinatorial algorithms to solve it. Many
combinatorial algorithms, like max-flow,
matching, min-cost flow, shortest path, spanning
tree, , can be derived within this framework.
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Approximation Algorithm
How do we adapt the primal-dual method for
approximation algorithms?
We want to construct a primal feasible solution x
and a dual feasible solution y so that cx and by
are close.
Avoid losing too much
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Approximate Optimality Condition
Avoid strict inequality
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Approximate Complementary Slackness Condition
Primal complementary slackness condition
Dual complementary slackness condition
Only a sufficient condition
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Vertex Cover
Primal complementary slackness condition
Dual complementary slackness condition
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Approximate Optimality Conditions
Primal complementary slackness condition
Just focus on this!
Pick only vertices that go tight.
Dual complementary slackness condition
Pick only edge with one vertex in the vertex
cover.
This is nothing!
This would imply a 2-approximation.
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Algorithm
Pick only vertices that go tight.
Algorithm (2-approximation for vertex
cover) Initially, x0, y0 When there is an
uncovered edge Pick an uncovered edge, and raise
y(e) until some vertices go tight. Add all tight
vertices to the vertex cover. Output the vertex
cover x.
This is the greedy matching 2-approximation when
every vertex has the same cost.
Familiar?
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Multicut
Given k source-sink pairs (s1,t1), (s2,t2),
...,(sk,tk), a multicut is a set of edges whose
removal disconnects each source-sink pair.
The multicut problem asks for the minimum weight
multicut.
Today we only consider a tree!
s2
t1
t2
s1
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Hardness
Why would this problem is hard?
Vertex cover.
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Primal Dual Programs
Primal complementary slackness condition
Dual complementary slackness condition
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Approximate Optimality Conditions
Primal complementary slackness condition
Only pick saturated edges.
Dual complementary slackness condition
two
At most one edge can be picked from a path
carrying nonzero flow.
Relaxed dual complementary slackness condition
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Algorithmic Idea
We want to construct flow and cut so that
Only pick saturated edges.
At most two edges can be picked from a path
carrying nonzero flow.
How to choose the flow?
We should avoid sending flow along a long path
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Algorithm
Multicut in Tree (2-approximation
algorithm) Initially, f0, D0 Flow
routing Find a pair (si,ti) with lowest lowest
common ancestor, Greedily send flow from si to
ti. Add all the saturated edges to D. Let e1,
e2,, el be the ordered list of edges in
D. Reverse delete In reverse order, delete an
edge e from D if D-e is still a multicut. Output
the flow and the cut.
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Reverse Delete
At most two edges can be picked from a path
carrying nonzero flow.
Lemma. After reverse delete, the above condition
is satisfied.
v
i can send a flow, so e is added after e.
e
u
there exists j where e is the only edge.
e
j is processed earlier than i.
Idea?
e
tj
sj
si
ti
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Bonus Question
Bouns Question 5 (50) Show that a basic
solution of the linear program for multicut in
tree has an edge of value at least 1/2.
Bouns Question 6 (50) Show that a basic
solution of the linear program for multiway cut
has an edge of value at least 1/2.
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Remarks

• Network design
• Facility location
• Online primal dual

A unify primal-dual method for exact and
approximation algorithm?