1 / 44

Lecture 23 Greedy Strategy

Lecture 23 Greedy Strategy. What is a submodular function?. Consider a function f on all subsets of a set E . f is submodular if. Set-Cover.

Download Presentation

Lecture 23 Greedy Strategy

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lecture 23 Greedy Strategy

  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. Theorem Weighted Connected Vertex-Cover has a (1+ln Δ)-approximation. This is the best-possible!!!

  31. End Thanks!

More Related