primal dual
DESCRIPTION
Tecnica primale duale per la risoluzione di problemi di ottimizzazione. Breve introduzioneTRANSCRIPT
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Carlo Lombardi, June 2008 Theoretical Computer Science
Primal-Dual Algorithms
A brief survey of Primal-Dual Algorithms
as an approximation technique for optimization problems
Scribe:Carlo [email protected]
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Carlo Lombardi, June 2008 Theoretical Computer Science 2
Overview
Introduction
The Minmum Weighted Vertex Cover Problem (WVC)
WVC as a ILP:
Solving WVC by rounding up a fractional solution
Solving WVC by Primal-Dual Strategy:
Duality: Background theoretic propertiesAlgorithmAnalysis
Example (on the blackboard)
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Introduction
• We have seen many algorithms based on Linear Program (LP), typically involving the following strategy:
•We arise the initial difficult of the problem by relaxing it•We sacrifice the optimal solution to find a good approximate solution by solving the relaxed problem
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Minimum Weighted Vertex Cover
Vertex Cover Problem“Each edge is covered by at least one node”
+Weighted Verteces
“Each vertex has a weight”
+Minimization of total weight“Minimize the total weight”
=
Minimum Weighted Vertex Cover (WVC)
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WVC: ILP and LP formulation
We formulate the WVC as an Integer Linear Program (ILP) defining a variable xi for each vertex (xi=1 if vertex i belongs to the cover, 0 otherwise).
ILP FORMULATION
LP FORMULATION
by relaxing integrality constraints
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WVC: Rounding the LP solution
Primal-Dual Method
We need to solve LP formulation…it can be
expensive for problems having many constraints!!!
Can we do something clever?
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A different approach to LP relaxations:Primal-Dual strategy
Main idea:
!!! Don’t solve LP totally !!!Obtain a feasible integral solution to the LP (Primal) from scratch using a related LP (Dual) to guide your decision.
!!! Don’t solve LP totally !!!Obtain a feasible integral solution to the LP (Primal) from scratch using a related LP (Dual) to guide your decision.
LP Primal
LP Dual
Good approximated solution
“Solve me”
“I’ll be your guide”
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P-D strategy: Background theoretic properties (1/2)
PRIMAL DUAL
(Weak Duality) For any feasible Primal-Dual solution pair (x,y):
= if (x,y) is optimal
(Strong Duality) If either the Primal or Dual have bounded optimal solution, the both of them do. Moreover, their objective functions values are qual. That is:
(Complementary Slackness) Let (x,y) be a solutions to a primal-dual pair of LPs with bounded optima. Then x and y are both optimal iff all of the following hold
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P-D strategy: Background theoretic properties (2/2)
(Weak Duality) For any feasible Primal-Dual solution pair (x,y):
The dual solution is a lover bound for primal solution
= if (x,y) is optimal
(Strong Duality) If either the Primal or Dual have bounded optimal solution, the both of them do. Moreover, their objective functions values are qual. That is:
At the optimum the evaluation of solutions coincides(Complementary Slackness) Let (x,y) be a solutions to a primal-dual pair of LPs with bounded optima. Then x and y are both optimal iff all of the following hold
Only If a dual constraints is tight the corresponding primal variables can be greater than 0 (it can participate to the primal solution)
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Primal-Dual strategy
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WVC : The D-P Algorithm
Primal Dual
1. Maintains an integer solution x of ILP and a feasible solution y for DLP
2. Examines x and y3. Derives a ‘more feasible’ solution
x and a ‘better’ solution y4. Ends when the integer solution
becomes feasible5. Evaluates the integer solution
comparing it with dual solution
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Analysis of Program 2.7
Note that for every it holds:
(1)
The o.f. is infact
From the (1)
Because we are considering all vertices in V
Each edge in E is taken two times
(Weak Duality)
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References
• G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti Spaccamela, M. Protasi, Complexity and Approximation, Springer, 1998, Chapter 2
•Michel X. Goemans, David P. Williamson, The primal dual method for approximations algorithms and its application to network design problems, PWS Publishing Co.,1997, Chapter 4
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“Follia è agire sempre allo stesso modo ed aspettarsi un risultato diverso”
(A.Einstien)
GrazieBuona Giornata