Lecture 23 Greedy Strategy

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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

What is a submodular function?

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

f is submodular if

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
• The earlier, the better!
• Monotone decreasing gain!
Theorem

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

The same result holds for 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
• 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 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
• Given a connected graph, find a minimum vertex-cover which induces a connected subgraph.
• 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.
|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)

-p-q
• -p-q is submodular.
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.
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.

Theorem Weighted Connected Vertex-Cover

has a (1+ln Δ)-approximation.

This is the best-possible!!!

End

Thanks!