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## Lecture 24 NP-Complete Problems

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**A < m B**p • A set A in Σ* is said to be polynomial-time many-one reducible to B in Γ* if there exists a polynomial-time computable function f: Σ* → Γ* such that x ε A iff f(x) ε B.**A = Hamiltonian cycle (HC)**• Given a graph G, does G contain a Hamiltonian cycle?**B = decision version of Traveling Salesman Problem (TSP)**• Given n cities and a distance table between these n cities, find a tour (starting from a city and come back to start point passing through each city exactly once) with minimum total length. • Given n cities, a distance table and k > 0, does there exist a tour with total length < k?**HC <m TSP**p • From a given graph G, we need to construct (n cities, a distance table, k).**SAT < m 3-SAT**p • SAT: Given a Boolean formula F, does F have a satisfied assignment? • An assignment is satisfied if it makes F =1. • 3-SAT: Given a 3-CNF F, does F have a satisfied assignment?**Property of < m**p p p p • A <m B and B <m C imply A <m C • A <m B and B ε P imply A ε P p**NP-complete**• A set A is NP-hard if for any B in NP, B <m A. • A set A is NP-complete if it is in NP and NP-hard. • A decision problem is NP-complete if its corresponding language is NP-complete. • An optimization problem is NP-hard if its decision version is NP-hard. p**(2) Cook Theorem**SAT is NP-complete**The last tape should contain**The final state.