1. Example Question on Linear Program, Dual and NP-Complete Proof COT5405 Spring 11

2. 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)

3. Integer Program Variables: – u ij = 1 if vertex i and j are connected in the resulted graph, otherwise u ij = 0. Note u ii = 1. – v i = 1 if vertex i is removed, otherwise v i = 0. The objective function and 2 constraints vivi vjvj u ij 110 100 010 001

4. Leftover Connectivity Consider node pairs i and j, where (i,j) is NOT an edge. i j h i j h UijUjhUhi 111 100 010 00X

5. Final Formulation

6. LP relaxation

7. Dual

8. |E| rows 1 row

9. Ready to write dual How many constraints in Primal – How many constraints in Dual – Dual variable – For the constraint on (i,j) belonging to E, define x ij – For the constraint on i,j,h belonging to V, define y ijh – For the constraint on the aggregate v i, define z

11. Final Dual

12. 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?

13. 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

14. 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?

15. 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

16. 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

17. Thank You Q & A