Transitivity of  poly Theorem: Let ,  ’, and  ’’ be three decision problems such that   poly  ’ and  ’  poly  ’’. Then  poly  ’’. Proof: Corollary: If ,  ’  NP such that  ’  poly  and  ’  NP-complete, then   NP-complete Proof: –How can we prove that   NP-hard? –How can we prove that   NP-complete?

Proving NP-Completeness SAT 3-CNF-SAT Subset-Sum CliqueHamiltonian CycleVertex-CoverTraveling Salesman

Example NP-Complete Problems 3-CNF-SAT –Input: Boolean formula f in CNF, such that each clause consists of exactly three literals. –Question: Is f satisfiable. Hamiltonian Cycle –Input: G = (V,E), undirected graph. –Does G have a cycle that visits each vertex exactly once (Hamiltonian Cycle)? Traveling Salesman –Input: A set of n cities with their intercity distances and an integer k. –Question: Does there exist a tour of length less than or equal to k? A tour is a cycle that visits each vertex exactly once.

Example 1 Show that the traveling salesman problem is NP-complete, assuming that the Hamiltonian cycle problem is NP-complete.

Example 2 Prove that the Problem Clique is NP- Complete. Proof: 1.Clique  NP 2.Clique  NP-Hard 3-CNF-SAT  poly Clique

Example 3 Prove that the problem Vertex Cover is NP- Complete Proof:

Example 4 Prove that the problem Independent Set is NP-Complete Proof:

Example NP-Complete Problems (Cont.) Subset Sum 3-Coloring 3D-Matching Hamiltonian Path Partition Knapsack Bin Packing Set Cover Multiprocessor Scheduling Longest Path …