Lecture 28 Approximation of Set Cover

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Presentation transcript:

Lecture 28 Approximation of Set Cover

Min Set Cover Red + Green

Greedy Algorithm

Observation

Theorem

Theorem Greedy Algorithm produces an approximation within ln n +1 from optimal. The same result holds for weighted set-cover.

Theorem Proof.

Hierarchy of Approximation

Theorem Proved using PCP system

MCDS Theorem (Guha-Khuller, 1998)

NP=P!

NP=P!

Connected Vertex-Cover Given a connected graph, find a minimum vertex-cover which induces a connected subgraph.

Theorem Connected Vertex-Cover has a 3-approximation.

Weighted Connected Vertex-Cover Given a vertex-weighted connected graph, find a connected vertex-cover with minimum total weight. Theorem Weighted Connected Vertex-Cover has a (1+ln Δ)-approximation where Δ is the maximum node degree of input graph. This is the best-possible!!!

Theorem

NP=P!

NP=P!