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This lecture examines approximation algorithms for set cover problems, focusing on the red and green greedy algorithms. The study proves that the greedy algorithm provides an approximation within ln n + 1 from the optimal solution and extends the result to weighted set cover problems. Additionally, it explores the Hierarchy of Approximation Theorem using the PCP system and MCDS Theorem. The lecture delves into the Connected Vertex-Cover problem, aiming to find a minimum vertex cover inducing a connected subgraph in a connected graph. The Weighted Connected Vertex-Cover problem for vertex-weighted connected graphs is also discussed, presenting a (1+ln Δ)-approximation where Δ is the maximum node degree in the input graph. The study concludes with the significant result that NP=P!
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Min Set Cover Red + Green
Theorem Greedy Algorithm produces an approximation within ln n +1 from optimal. The same result holds for weighted set-cover.
Theorem Proof.
Theorem Proved using PCP system
MCDS Theorem (Guha-Khuller, 1998)
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. TheoremWeighted Connected Vertex-Cover has a (1+ln Δ)-approximation where Δ is the maximum node degree of input graph. This is the best-possible!!!
