Lecture 23 Greedy Strategy
What is a submodular function?
Consider a function f on all subsets of a set E.
f is submodular if
f ( A)  f ( B)  f ( A  B)  f ( A  B)
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’ .
Example of Submodular Function
For a subcollection A of C , define
f ( A) | s  A S|.
Then
f ( A)  f ( B)  f ( A  B)  f ( A  B)
Greedy Algorithm
C '  ;
while | E | f (C ') do
choose S  C to maximize f (C ' {S}) and
C '  C ' {S};
Analysis
Suppose S 1, S 2, ..., Sk are selected by Greedy
Algorithm. Denote Ci  {S 1, ..., Si}. Then
f (Ci  1)  f (Ci)  (| E |  f (Ci)) / opt
(| E |  f (Ci ))(1  1 / opt )  | E |  f (Ci  1)
| E |  f (Ci  1)  (| E |  f (Ci ))(1  1 / opt )
 (| E |  f (Ci  1))(1  1 / opt )

 | E | (1  1 / opt )
i 1
Choose i to be the largest one satisfying
opt  | E |  f (Ci ).
Then
k  i  opt
opt  | E | (1  1 / opt )
i
2
opt  | E | (1  1 / opt )
| E |e
So
i / opt
i
(note : 1  x  e )
i  opt ln (| E | / opt )
Thus,
k  opt  i
 opt (1  ln (| E | / opt ))
x
Analysis
Suppose S 1, S 2, ..., Sk are selected by Greedy
Algorithm. Denote Ci  {S 1, ..., Si}. Then
f (Ci  1)  f (Ci)  (| E |  f (Ci)) / opt
Denote Xf ( A)  f ( A  { X })  f ( A).
Let C*  { X 1, X 2,..., Xopt} be an optimal solution.
Denote Cj*  { X 1,..., Xj}.
By greedy rule,
Si  1 f (Ci )  Xj  1 f (Ci )
for all 0  j  opt  1
Thus, Si  1 f (Ci )  (0 j opt1 Xj  1 f (Ci )) / opt
 (0 j opt1 Xj  1 f (Ci  Cj*)) / opt
 ( f (Ci  C*)  f (Ci )) / opt
 (| E |  f (Ci )) / opt
What’s we need?
A  B  Xf ( A)  Xf ( B)
Actually, this inequality holds if and only if f is
submodular and
A  B  f ( A)  f ( B)
(monotone increasing)
A  B  xf ( A)  xf ( B )
if and only if f is submodular and
monotone increasing
( f is submodular ) implies
A  B  xf ( A)  xf ( B) for x  B
( f is monotone increasing ) implies
A  B  xf ( A)  xf ( B) for x  B
f ( A)  f ( B )  f ( A  B )  f ( A  B )
f ( A)  f ( A  B )  f ( A  B )  f ( B )
Denote C  A \ B  A  A  B  A  B  B.
Then
Cf ( A  B )  Cf ( B )
Suppose f is submodular . For U  V and x  V ,
set A  U  {x} and B  V . Then
A \ B  {x}, A  B  U , and hence
xf (U )  xf (V ).
Conversely , suppose above inequality holds for all
U  V and x  V . Let C  { y1,..., yk} and denote
Ci  { y1,..., yi}.
Cf ( A  B)  y1 f ( A  B )  y 2 f (( A  B )  C1)
   ykf (( A  B )  Ck  1)
Cf ( B)  y1 f ( B )  y 2 f ( B  C1)
   ykf ( B  Ck  1)
For A  B and x  B,
xf ( A)  xf ( B)  0
if and only if
f ( A  {x})  f ( A),
i.e., f is 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 setcover.
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’ .
Greedy Algorithm
C '  ;
while | E | f (C ' ) do
choose S  C to maximize Sf (C ' ) / w( S ) and
C '  C '{S };
output C '.
A General Problem
Consider a monotone increasing , submodular
function f defined on all subsets of a set E.
Define ( f )  { A | x  E , xf ( A)  0}.
min c( A)   xA c( x)
s.t.
A  ( f )
Greedy Algorithm
A  ;
while f *  f ( A) do
choose x  E to maximize xf ( A) / c( x) and
A  A  {x};
output A.
( f *  max f ( A)  f ( A) for any A  ( f ))
A General Theorem
If f ()  0, f is an integer function, and c( x)  0
for x  E , then Greedy gives a (1  ln  ) - approximat ion
where   max xE f ({x}).
Remark: If f ()  0, we may consider
g ( A)  f ( A)  f ().
Therefore   max xE ( f ({x})  f ()).
Proof
Suppose x1, x 2, ..., xk are selected by Greedy Algorithm in
the order of their appearance . Denote Ai  {x1, ..., xi}
and ri  xif ( Ai  1).
Suppose A*  { y1, ..., yh} is an optimal solution. For each
ye  A*, denote zei  ye f ( Ai  1), ci  c( xi ) and
k 1
ci
ck
w( ye)   ( zei  ze , i  1)  zek
ri
rk
i 1
k
c1
ci ci  1
 ze1   ( 
) zei
r1
ri  1
i  2 ri
By Greedy rule and submodular property of f ,
r1 r 2
rk
 
c1 c 2
ck
**************************
r1 x 2 f ( A0)

by Greedy rule, where A0  
c1
c2
x 2 f ( A1) r 2


since f is submodular
c2
c2
k
r   z
j
j i
k
r
j
ei
for i  1, ..., k
ye A*
 f ( Ak )  f ( Ai  1)  f ( Ai  1  A*)  f ( Ai  1)
j i
h
  yjf ( Ai  1  A * j  1)
j 1
h
  yjf ( Ai  1) by submodular and monotone
j 1
increasing properties of f
h
  zji
j 1
k
ri
c( Ak )   ci
i 1 ri
k 1
k
k
k
ci
ck
  (  rj   rj )   rj
rk j  k
i 1 ri j i
j i 1
k
c1 k
ci ci  1 k
  rj   ( 
)  rj
r1 j 1
ri  1 j  i
i  2 ri
k
c1
ci ci  1
  ze1   ( 
)  zei
r1 yeA*
ri  1 yeA*
i  2 ri

 w( y )
e
ye A*
ci c( ye)
By greedy rule,

.
ri
zei
By the submodular and monotone increasing property
of f ,
zei  ze, i  1.
Therefore,
k 1
ci
ck
w( ye)   ( zei  ze , i  1)  zek
ri
rk
i 1
k 1
c( ye)
c( ye)
  ( zei  ze , i  1)
 zek
zei
zek
i 1
ze 1
1
 c( ye)
i 1 i
since zei are integers.
1
2
3
zek
Ze2
ze1
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, 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)


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 3approximation.
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!