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 Lombardilombardi.carlo@gmail.com

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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