Carlo Lombardi,  June  2008 Theoretical  Computer  Science Primal-Dual Algorithms A brief survey of  Primal-Dual Algorithm...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science Overview <ul><li>Introduction </li></ul><ul><li>The Minmum Weig...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science Introduction <ul><li>We have seen many algorithms based on  Lin...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science Minimum Weighted Vertex Cover Vertex Cover Problem “ Each edge ...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science WVC: ILP and LP formulation We formulate the WVC as an Integer ...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science WVC: Rounding the LP solution Primal-Dual Method We need to  so...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science A different approach to LP relaxations: Primal-Dual strategy Ma...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science P-D strategy: Background theoretic properties (1/2) PRIMAL DUAL...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science P-D strategy: Background theoretic properties (2/2) (Weak Duali...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science Primal-Dual strategy
Carlo Lombardi,  June  2008 Theoretical  Computer  Science WVC : The D-P Algorithm <ul><li>Maintains an integer solution  ...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science Analysis of Program 2.7 (Weak Duality) Note that for every  it ...
Carlo Lombardi,  June  2008 Theoretical  Computer  Science References <ul><li>G. Ausiello, P. Crescenzi, G. Gambosi, V. Ka...
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Primal Dual

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Primal Dual

  1. 1. 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]
  2. 2. 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>
  3. 3. 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>
  4. 4. 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)
  5. 5. 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
  6. 6. 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?
  7. 7. 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”
  8. 8. 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
  9. 9. 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)
  10. 10. Carlo Lombardi, June 2008 Theoretical Computer Science Primal-Dual strategy
  11. 11. 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
  12. 12. 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
  13. 13. 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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