Approximation Algorithms

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Approximation Algorithms

• Duality
• My T. Thai _at_ UF

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Duality

• Given a primal problem
• P min cTx subject to Ax b, x 0
• The dual is
• D max bTy subject to ATy c, y 0

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An Example
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Weak Duality Theorem

• Weak duality Theorem
• Let x and y be the feasible solutions for P and
  D respectively, then
• Proof Follows immediately from the constraints

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Weak Duality Theorem

• This theorem is very useful
• Suppose there is a feasible solution y to D. Then
  any feasible solution of P has value lower
  bounded by bTy. This means that if P has a
  feasible solution, then it has an optimal
  solution
• Reversing argument is also true
• Therefore, if both P and D have feasible
  solutions, then both must have an optimal
  solution.

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Hidden Message

Strong Duality Theorem If the primal P has an
optimal solution x then the dual D has an
optimal solution y such that cTx bTy
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Complementary Slackness

• Theorem
• Let x and y be primal and dual feasible solutions
  
• respectively. Then x and y are both optimal iff
  two
• of the following conditions are satisfied
• (ATy c)j xj 0 for all j 1n
• (Ax b)i yi 0 for all i 1m

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Proof of Complementary Slackness

• Proof
• As in the proof of the weak duality theorem, we
• have cTx (ATy)Tx yTAx yTb (1)
• From the strong duality theorem, we have

(2) (3)
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Proof (cont)

• Note that
• and
• We have
• x and y optimal ? (2) and (3) hold
• ? both sums (4) and (5) are zero
• ? all terms in both sums are zero (?)
• ? Complementary slackness holds

(4)
(5)
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Why do we care?

• Its an easy way to check whether a pair of
  primal/dual feasible solutions are optimal
• Given one optimal solution, complementary
  slackness makes it easy to find the optimal
  solution of the dual problem
• May provide a simpler way to solve the primal

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Some examples

• Solve this system

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Min-Max Relations

• What is a role of LP-duality
• Max-flow and Min-Cut

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Max Flow in a Network

• Definition Given a directed graph G(V,E) with
  two distinguished nodes, source s and sink t, a
  positive capacity function c E ? R, find the
  maximum amount of flow that can be sent from s to
  t, subject to
• Capacity constraint for each arc (i,j), the flow
  sent through (i,j), fij bounded by its capacity
  cij
• Flow conservation at each node i, other than s
  and t, the total flow into i should equal to the
  total flow out of i

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An Example
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Formulate Max Flow as an LP

• Capacity constraints 0 fij cij for all (i,j)
• Conservation constraints
• We have the following

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LP Formulation (cont)
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LP Formulation (cont)
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Min Cut

• Capacity of any s-t cut is an upper bound on any
  feasible flow
• If the capacity of an s-t cut is equal to the
  value of a maximum flow, then that cut is a
  minimum cut

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Max Flow and Min Cut
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Solutions of IP

• Consider
• Let (d,p) be the optimal solution to this IP.
  Then
• ps 1 and pt 0. So define X pi pi 1
  and
• X pi pi 0. Then we can find the s-t cut
• dij 1. So for i in X and j in X, define dij
  1, otherwise dij 0.
• Then the object function is equal to the minimum
  s-t cut

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LP-relaxation

• Relax the integrality constraints of the previous
  IP, we will obtain the previous dual.

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Design Techniques

• Many combinatorial optimization problems can be
  stated as IP
• Using LP-relaxation techniques, we obtain LP
• The feasible solutions of the LP-relaxation is a
  factional solution to the original. However, we
  are interested in finding a near-optimal integral
  solution
• Rounding Techniques
• Primal-dual Schema

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Rounding Techniques

• Solve the LP and convert the obtained fractional
  solution to an integral solution
• Deterministic
• Probabilistic (randomized rounding)

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Primal-Dual Schema

• An integral solution of LP-relaxation and a
  feasible solution to the dual program are
  constructed iteratively
• Any feasible solution of the dual also provides
  the lower bound of OPT
• Comparing the two solutions will establish the
  approximation guarantee