Example Question on
Linear Program, Dual and
NP-Complete Proof
COT5405 Spring 11
Question
• Given an undirected connected graph G = (V,E)
and a positive integer k ≤ |V|.
• Two vertices u and v are connected if and only
if there exists at least one path from u to v.
• For all the possible vertex pairs, we want to
remove k vertices from G, so that
• the number of connected vertex pairs in the
resulting graph is minimized.
• We call it k-CNP (critical node problem)
Integer Program
• Variables:
– uij = 1 if vertex i and j are connected in the
resulted graph, otherwise uij = 0. Note uii = 1.
– vi = 1 if vertex i is removed, otherwise vi = 0.
• The objective function and 2 constraints
1
min  uij
2 i , jV
(i, j )  E , ui j  vi  v j  1
v
iV
i
k
vi
vj
uij
1
1
0
1
0
0
0
1
0
0
0
1
Leftover Connectivity
• Consider node pairs i and j, where (i,j) is NOT
an edge.
uij  u jh  uhi  1, (i, j , h)  V
h
i
j
Uij
Ujh Uhi
1
1
1
1
0
0
0
1
0
0
0
X
h
i
j
Final Formulation
LP relaxation
Dual
Min
c X
j
Max
j
j
s.t.
a X
ij
yb
ya
i i
i
j

bi
for all i
j
Xj 
s.t.
i ij
 c j for all j
i
0
for all j
yi
 0
for all i
Min
1 1 1 1
( , , ,
2 2 2 2
1
)(u12u13
2
un 1, n ) '
n
 
 2
s.t.


1
0




 1 1
 1 1

 1 1





0
0



n
 
 2
0
0
n
0 0 0 0 0 01
|E| rows
1
0
1 0 0 0 0 0 0 0
0
0
1
0 0 0 0 0 00
0
0
1 0 0 0 0 0 0 0
0
0
n
3   rows
 3
0
0
0 0 0 0 0 0 1 1 1
1 row

 u 
  12   1 
  u13   
1

  

  
  u23   
0
1

  
  1 
0 
  un 1,n    
0 
  1 
v
 1   

  

  
v
 n   

  

  k 




Ready to write dual
• How many constraints in Primal –
n
• How many constraints in Dual –  2 
n
(3
• Dual variable  3   | E | 1)dimension in total
n
3    | E | 1
 3
– For the constraint on (i,j) belonging to E, define xij
– For the constraint on i,j,h belonging to V, define
yijh
– For the constraint on the aggregate vi, define z
Max
(1,1,
, 1, 1, 1 ,  k )( x12
y123
z) '
n
3  | E |1
 3
s.t.


1

0



0



1
1

0
0



n
3 
 3
|E|
1 1 1
1 1 1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1

2
 x   
0   12   1 
 x
 
0   13   2 



  

 1

 2

  
  y123    1 

  
  y312   2 
1
1
 y132   
1 
 2

  
1 

  
1 
 0
 z   

0

0
 
Final Dual
NP-Complete Proof
• Decision Version
– Given an undirected connected graph G and
positive integer k
– a value L<n(n-1)/2
– is there a set of k vertices, whose removal makes
the number of connected vertex pairs in the
resulting graph is at most L?
In NP
• Given such a set of k vertices,
• Remove them from the graph,
• Calculate the number of connected pairs using
DFS or BFS in polynomial time,
• Compare with L – Give answer: Yes or No
Is NP-hard
Reduction from Vertex Cover (VC)
• Instance of VC: given a graph G = (V,E) where
|V|= n, is there a vertex cover of size at most
k?
• Instance of k-CNP: on the graph G, is there a
set of k vertices whose removal makes the # of
connected pairs 0?
Is NP-hard
• Forward: If we can have a VC of size k --->
delete those k nodes ---> connectivity = 0
• Backward: If we can delete k nodes to make #
connections 0 --> no edges left -> vertex cover
of size k
NP-Completeness
• In NP
• NP-hard
• For an alternative proof, please refer to
A. Arulselvan et al, ``Detecting Critical Nodes In Sparse
Graphs’’, J. Computers and Operations Research, 2009.
http://plaza.ufl.edu/clayton8/cnp.pdf
Thank You
Q&A