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Unformatted text preview: Introduction to Algorithms December 8, 2006 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Madhu Sudan Problem Set 9 Solutions Problem Set 9 Solutions Problem 9-1. Subgraph Isomorphism Two graphs G 1 = ( V 1 ,E 1 ) and G 2 = ( V 2 ,E 2 ) are said to be isomorphic if there exists a bijection π : V 1 → V 2 , providing a one-to-one correspondence between the vertices of the two graphs, such that ( u,v ) ∈ E 1 if and only if ( π ( u ) ,π ( v )) ∈ E 2 . A graph H = ( V H ,E H ) is a subgraph of G = ( V G ,E G ) if V H ⊆ V G and E H ⊆ E G . Consider the subgraph-isomorphism problem : given two graphs G and H , determine whether G is isomorphic to a subgraph of H . Prove that the subgraph-isomorphism problem is NP-complete. Solution: First, we argue that the subgraph-isomorphism problem belongs to NP. The verification algorithm for subgraph isomorphism, V SUB , takes in a graph G = ( V g ,E g ) , the graph H = ( V h ,G h ) , the vertices of the corresponding subgraph J ⊆ V h , and a permutation function π : J → V g . Given J and π , we can easily verify in polynomial time whether G is isomorphic to the subgraph of H determined by J . For every edge ( u,v ) ∈ E g , search E h to see if ( π ( u ) ,π ( v )) ∈ E h , and search J to see if π ( u ) ∈ J and π ( v ) ∈ J . If all searches returns true for all edges ( u,v ) , then G is isomorphic to a subgraph of H with J as its vertices....
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- Fall '08
- Algorithms, Computational complexity theory, Clique, NP-complete, linear difference constraints