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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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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’ .

slide12

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’ .

slide31

1

2

3

slide32

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.
slide34
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.
slide35
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.
slide38
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.
e p a
|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)

slide40
-p-q
  • -p-q is submodular.
theorem1
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.
theorem2
Theorem
  • Connected Vertex-Cover has a 3-approximation.
weighted connected vertex cover
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!!!

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End

Thanks!

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