Chapter 2. Greedy Strategy
II. Submodular function
Ding-Zhu Du
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 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 totalweight 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
ck
ci
w( ye)   ( zei  ze, i  1)  zek
rk
ri
i 1
k
ci ci  1
c1
) zei
 ze1   ( 
ri  1
r1
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
for i  1, ..., k
ei
j i
ye A*
k
r  f (A )  f (A
j
k
)  f ( Ai  1  A*)  f ( Ai  1)
i 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
ci k
ck k
  ( 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!!!
Thanks, End