L11 - Approximation algorithms (Ch.35) Polynomial-time...

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Unformatted text preview: Approximation algorithms (Ch.35) Polynomial-time algorithms that always return near-optimal solutions An example The vertex-cover (VC) problem A vertex cover of an undirected graph G= ( V , E ) is a subset such that for every edge ( u,v ) of G , either Objective: find a minimum vertex cover, i.e., a vertex cover with minimum size We are almost sure that the VC problem cannot be solved in polynomial time . '. or ' V v V u V V ' a Vertex Cover A simple algorithm An example Theorem: The size of the VC returns by APPROX-VERTEX-COVER is at most twice of the size of the optimal (smallest) VC. That is C: VC returned by APPROX-VERTEX-COVER C*: optimal VC |C| 2|C*| Theorem: The size of the VC returns by APPROX-VERTEX-COVER is at most twice of the size of the optimal (smallest) VC. Proof. Let E f be the set of edges picked by the algorithm during some iterations. Then, { } . , ) , ( v u C f E v u v = E f = {(b,c),(e,f),(d,g)}. C={b,c,e,f,d,g} Theorem: The size of the VC returns by APPROX-VERTEX-COVER is at most twice of the size of the optimal (smallest) VC. Proof. Let E f be the set of edges picked by the algorithm during Theorem: The size of the VC returns by APPROX-VERTEX-COVER is at most twice of the size of the optimal (smallest) VC....
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L11 - Approximation algorithms (Ch.35) Polynomial-time...

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