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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 Lombardi [email_address]
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Carlo Lombardi, June 2008 Theoretical Computer Science Overview <ul><li>Introduction </li></ul><ul><li>The Minmum Weighted Vertex Cover Problem (WVC) </li></ul><ul><li>WVC as a ILP: </li></ul><ul><ul><li>Solving WVC by rounding up a fractional solution </li></ul></ul><ul><ul><li>Solving WVC by Primal-Dual Strategy: </li></ul></ul><ul><ul><ul><li>Duality: Background theoretic properties </li></ul></ul></ul><ul><ul><ul><li>Algorithm </li></ul></ul></ul><ul><ul><ul><li>Analysis </li></ul></ul></ul><ul><li>Example (on the blackboard) </li></ul>
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Carlo Lombardi, June 2008 Theoretical Computer Science Introduction <ul><li>We have seen many algorithms based on Linear Program (LP) , typically involving the following strategy: </li></ul><ul><li>We arise the initial difficult of the problem by relaxing it </li></ul><ul><li>We sacrifice the optimal solution to find a good approximate solution by solving the relaxed problem </li></ul>
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Carlo Lombardi, June 2008 Theoretical Computer Science 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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Carlo Lombardi, June 2008 Theoretical Computer Science WVC: ILP and LP formulation We formulate the WVC as an Integer Linear Program (ILP) defining a variable x i for each vertex ( x i =1 if vertex i belongs to the cover, 0 otherwise). ILP FORMULATION LP FORMULATION by relaxing integrality constraints
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Carlo Lombardi, June 2008 Theoretical Computer Science 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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Carlo Lombardi, June 2008 Theoretical Computer Science 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 . LP Primal LP Dual Good approximated solution “ Solve me” “ I’ll be your guide”
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Carlo Lombardi, June 2008 Theoretical Computer Science 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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Carlo Lombardi, June 2008 Theoretical Computer Science 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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Carlo Lombardi, June 2008 Theoretical Computer Science Primal-Dual strategy
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Carlo Lombardi, June 2008 Theoretical Computer Science WVC : The D-P Algorithm <ul><li>Maintains an integer solution x of ILP and a feasible solution y for DLP </li></ul><ul><li>Examines x and y </li></ul><ul><li>Derives a ‘more feasible’ solution x and a ‘better’ solution y </li></ul><ul><li>Ends when the integer solution becomes feasible </li></ul><ul><li>Evaluates the integer solution comparing it with dual solution </li></ul>Primal Dual
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Carlo Lombardi, June 2008 Theoretical Computer Science Analysis of Program 2.7 (Weak Duality) 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
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Carlo Lombardi, June 2008 Theoretical Computer Science References <ul><li>G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti Spaccamela, M. Protasi, Complexity and Approximation , Springer, 1998 , Chapter 2 </li></ul><ul><li>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 </li></ul>
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