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# Lecture28 - Populating the NP-Completeness Universe CSE 421...

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1 CSE 421 Algorithms Richard Anderson Lecture 28 NP-Completeness Populating the NP-Completeness Universe Circuit Sat < P 3-SAT 3-SAT < P Independent Set 3-SAT < P Vertex Cover Independent Set < P Clique 3-SAT < P Hamiltonian Circuit Hamiltonian Circuit < P Traveling Salesman 3-SAT < P Integer Linear Programming 3-SAT < P Graph Coloring 3-SAT < P Subset Sum Subset Sum < P Scheduling with Release times and deadlines Cook’s Theorem The Circuit Satisfiability Problem is NP- Complete Circuit Satisfiability – Given a boolean circuit, determine if there is an assignment of boolean values to the input to make the output true Circuit SAT AND OR AND AND OR OR AND NOT OR NOT x 1 x 2 x 3 x 4 x 5 AND AND NOT NOT AND OR NOT AND OR AND Find a satisfying assignment Proof of Cook’s Theorem Reduce an arbitrary problem Y in NP to X Let A be a non-deterministic polynomial time algorithm for Y Convert A to a circuit, so that Y is a Yes

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