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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.

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Lecture 23 greedy strategy

Lecture 23 Greedy Strategy


What is a submodular function
What is a submodular function?

Consider a function f on all subsets of a set E.

f is submodular if


Set cover
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’ .







Actually, this inequality holds if and only if f is submodular and

(monotone increasing)


Meaning of submodular
Meaning of Submodular

  • The earlier, the better!

  • Monotone decreasing gain!


Theorem
Theorem

Greedy Algorithm produces an approximation within ln n +1 from optimal.

The same result holds for weighted set-cover.


Weighted set cover
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’ .







1

2

3


ze1

zek

Ze2


Subset interconnection design
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.


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.


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.


Potential function r 1 r m
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


Connected vertex cover
Connected Vertex-Cover

  • Given a connected graph, find a minimum vertex-cover which induces a connected subgraph.



E p a
|E|-p(A) covered by 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)


-p-q covered by A.

  • -p-q is submodular.


Theorem1
Theorem covered by A.

  • 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.


Theorem2
Theorem covered by A.

  • Connected Vertex-Cover has a 3-approximation.


Weighted connected vertex cover
Weighted Connected Vertex-Cover covered by A.

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!!!


End covered by A.

Thanks!


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