The Primal-Dual Method for Approximation Algorithms PDF
Preview The Primal-Dual Method for Approximation Algorithms
The Primal-Dual Method for Approximation Algorithms Jochen Könemann University of Waterloo JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.1/44 What is the primal-dual method? lWhatistheprimal-dualmethod? n Originally proposed by Dantzig, Ford, and Fulkerson in 1956 as an alternate method to solve linear programs exactly Primal-Dual:FirstSteps Primal-Dual:HittingSets Primal-Dual:SteinerTrees PrimalDual:MCF JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.2/44 What is the primal-dual method? lWhatistheprimal-dualmethod? n Originally proposed by Dantzig, Ford, and Fulkerson in 1956 as an alternate method to solve linear programs exactly Primal-Dual:FirstSteps Primal-Dual:HittingSets n Method did not survive... but: Revised version of it has Primal-Dual:SteinerTrees become immensely popular for solving combinatorial PrimalDual:MCF optimization problems. JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.2/44 What is the primal-dual method? lWhatistheprimal-dualmethod? n Originally proposed by Dantzig, Ford, and Fulkerson in 1956 as an alternate method to solve linear programs exactly Primal-Dual:FirstSteps Primal-Dual:HittingSets n Method did not survive... but: Revised version of it has Primal-Dual:SteinerTrees become immensely popular for solving combinatorial PrimalDual:MCF optimization problems. Examples: Dijkstra’s shortest path algorithm, Ford and Fulkerson’s network flow algorithm, Edmond’s non-bipartite matching method, Kuhn’s assignment algorithm, ... JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.2/44 What is the primal-dual method? lWhatistheprimal-dualmethod? n Originally proposed by Dantzig, Ford, and Fulkerson in 1956 as an alternate method to solve linear programs exactly Primal-Dual:FirstSteps Primal-Dual:HittingSets n Method did not survive... but: Revised version of it has Primal-Dual:SteinerTrees become immensely popular for solving combinatorial PrimalDual:MCF optimization problems. Examples: Dijkstra’s shortest path algorithm, Ford and Fulkerson’s network flow algorithm, Edmond’s non-bipartite matching method, Kuhn’s assignment algorithm, ... Main feature: Reduce weighted optimization problems to easier unweighted ones. JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.2/44 What is the primal-dual method? P lWhatistheprimal-dualmethod? n All of the previous problems are in . Can we extend this NP method to -hard problems? Primal-Dual:FirstSteps Primal-Dual:HittingSets Primal-Dual:SteinerTrees PrimalDual:MCF JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.3/44 What is the primal-dual method? P lWhatistheprimal-dualmethod? n All of the previous problems are in . Can we extend this NP method to -hard problems? Primal-Dual:FirstSteps Primal-Dual:HittingSets Yes! Bar-Yehuda and Even use this in 1981 in their Primal-Dual:SteinerTrees approximation algorithm for vertex-cover. PrimalDual:MCF JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.3/44 What is the primal-dual method? P lWhatistheprimal-dualmethod? n All of the previous problems are in . Can we extend this NP method to -hard problems? Primal-Dual:FirstSteps Primal-Dual:HittingSets Yes! Bar-Yehuda and Even use this in 1981 in their Primal-Dual:SteinerTrees approximation algorithm for vertex-cover. PrimalDual:MCF n Goemans and Williamson formalize this approach in 1992. Result is a general toolkit for developing approximation NP algorithms for -hard optimization problems. n The last 10 years have seen literally hundreds of papers that use the primal-dual framework. JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.3/44 lWhatistheprimal-dualmethod? Primal-Dual:FirstSteps lAnexample:VertexCover lAnILPforVertexCover lDualLP lPD:MainIdeas lVertex-cover: APD-Algorithm Primal-Dual: First Steps lPetersengraphexample lApproximationalgorithms Primal-Dual:HittingSets Primal-Dual:SteinerTrees PrimalDual:MCF JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.4/44 n Here’s a vertex cover of size 6. An example: Vertex Cover 5 lWhatistheprimal-dualmethod? Primal-Dual:FirstSteps lAnexample:VertexCover 0 lAnILPforVertexCover lDualLP lPD:MainIdeas 9 6 4 1 lVertex-cover: APD-Algorithm lPetersengraphexample lApproximationalgorithms Primal-Dual:HittingSets Primal-Dual:SteinerTrees 3 2 PrimalDual:MCF 7 8 n Goal: Find a minimum subset of the vertices C such that e \ C 6= ; for all edges e. JochenKönemann,September25,2004 GroupStrategyproofMechanismsforSteinerForests-p.5/44
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