NP-Complete problems NP-Complete problems - Ace

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✬✩SAT is NP-CompleteProof:• Step 1: SAT ∈ NP.Certificate is the “truth assignment”. Algorithm merelyhas to verify, in polynomial time, that the truthassignment produces TRUE.• Step 2: SAT ∈ NP-hard.by proving CKT-SAT ≤ p SAT.CS404/504✫Design and Analysis of Algorithms: Lecture 26Computer Science16✪
✬✩SAT: Poly-time reductionThe reduction is as follows:• For each wire x i in the circuit C, the formula φ has avariable x i .• For each gate in C, make a formula involving the variablesof its incident wires that fully describes the behaviour of thegate.– For example, the operation of the output OR gate(figure on the next page) is x 5 ↔ (x 1 ∨x 2 ).• The formula φ produced by the reduction is the AND of thecircuit-output variable with the conjunction of clausesdescribing the operation of each gate.CS404/504✫Design and Analysis of Algorithms: Lecture 26Computer Science17✪
Page 1 and 2: ✬✩NP-Complete problemsNP-comple
Page 3 and 4: ✬✩From P to NPC• However, the
Page 5 and 6: ✬✩Polynomial reductionsCS404/50
Page 7 and 8: ✬✩Implication of A ≤ p B• f
Page 9 and 10: ✬✩NP-hard and NP-Complete• De
Page 11 and 12: ✬✩CKT-SAT problemDecision probl
Page 13 and 14: ✬✩Alternative definition of NP-
Page 15: ✬✩Another NPC problem: SATFormu
Page 19: ✬✩SAT: Poly-time reduction• E

NP-Complete problems NP-Complete problems - Ace

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