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Chapter 2. Greedy Strategy

Chapter 2. Greedy Strategy. Ding-Zhu Du. II. Submodular function. What is a submodular function?. Consider a function f on all subsets of a set E . f is submodular if. Set-Cover.

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Chapter 2. Greedy Strategy

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  1. Chapter 2. Greedy Strategy Ding-Zhu Du II. Submodular function

  2. What is a submodular function? Consider a function f on all subsets of a set E. f is submodular if

  3. Set-Cover Given a collection C of subsets of a set E, find a minimum subcollection C’ of C such that every element of E appears in a subset in C’ .

  4. Example of Submodular Function

  5. Greedy Algorithm

  6. Analysis

  7. Analysis

  8. What’s we need?

  9. Actually, this inequality holds if and only if f is submodular and (monotone increasing)

  10. Meaning of Submodular • The earlier, the better! • Monotone decreasing gain!

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

  12. Weighted Set Cover Given a collection C of subsets of a set E and a weight function w on C, find a minimum total-weight subcollection C’ of C such that every element of E appears in a subset in C’ .

  13. Greedy Algorithm

  14. A General Problem

  15. Greedy Algorithm

  16. A General Theorem Remark:

  17. Proof

  18. 1 2 3

  19. ze1 zek Ze2

  20. Subset Interconnection Design • Given m subsets X1, …, Xm of set X, find a graph G with vertex set X and minimum number of edges such that for every i=1, …, m, the subgraph G[Xi] induced by Xi is connected.

  21. fi For any edge set E, define fi(E) to be the number of connected components of the subgraph of (X,E), induced by Xi. • Function -fi is submodular.

  22. Rank • All acyclic subgraphs form a matroid. • The rank of a subgraph is the cardinality of a maximum independent subset of edges in the subgraph. • Let Ei = {(u,v) in E | u, v in Xi}. • Rank ri(E)=ri(Ei)=|Xi|-fi(E). • Rank ri is sumodular.

  23. Potential Function r1+ּּּ+rm Theorem Subset Interconnection Design has a (1+ln m)-approximation. r1(Φ)+ּּּ+rm(Φ)=0 r1(e)+ּּּ+rm(e)<m for any edge

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

  25. For any vertex subset A, p(A) is the number of edges not covered by A. • For any vertex subset A, q(A) is the number of connected component of the subgraph induced by A. • -p is submodular. • -q is not submodular.

  26. |E|-p(A) • p(A)=|E|-p(A) is # of edges covered by A. • p(A)+p(B)-p(A U B) = # of edges covered by both A and B > p(A ∩ B)

  27. -p-q • -p-q is submodular.

  28. Theorem • Connected Vertex-Cover has a (1+ln Δ)-approximation. • -p(Φ)=-|E|, -q(Φ)=0. • |E|-p(x)-q(x) <Δ-1 • Δ is the maximum degree.

  29. Theorem • Connected Vertex-Cover has a 3-approximation.

  30. 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. This is the best-possible!!!

  31. Thanks, End

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