Chapter 2' Greedy Strategy

1
Chapter 2. Greedy Strategy

• II. Submodular function

Ding-Zhu Du
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
(No Transcript)
8
(No Transcript)
9
Analysis
10
(No Transcript)
11
Whats we need?
12

• Actually, this inequality holds if and only if f
  is submodular and
• (monotone increasing)

13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
Meaning of Submodular

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

19
Theorem

• Greedy Algorithm produces an approximation within
  ln n 1 from optimal.
• The same result holds for weighted set-cover.

20
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 .

21
Greedy Algorithm
22
A General Problem
23
Greedy Algorithm
24
A General Theorem
Remark
25
Proof
26
(No Transcript)
27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30
(No Transcript)
31
1
2
3
32
ze1
zek
Ze2
33
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 i1, , m, the subgraph GXi
  induced by Xi is connected.

34
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.

35
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.

36
Potential Function r1???rm

• Theorem Subset Interconnection Design has a
  (1ln m)-approximation.
• r1(F)???rm(F)0
• r1(e)???rm(e)ltm for any edge

37
Connected Vertex-Cover

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

38

• 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.

39
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
• gt p(A n B)

40
-p-q

• -p-q is submodular.

41
Theorem

• Connected Vertex-Cover has a (1ln
  ?)-approximation.
• -p(F)-E, -q(F)0.
• E-p(x)-q(x) lt ?-1
• ? is the maximum degree.

42
Theorem

• Connected Vertex-Cover has a 3-approximation.

43
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 (1ln ?)-approximation.
• This is the best-possible!!!

44
Thanks, End