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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’ .







Lecture 23 greedy strategy

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’ .







Lecture 23 greedy strategy

1

2

3


Lecture 23 greedy strategy

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.


Lecture 23 greedy strategy
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.


Lecture 23 greedy strategy
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.


Lecture 23 greedy strategy


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)


Lecture 23 greedy strategy
-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!!!


Lecture 23 greedy strategy
End covered by A.

Thanks!