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Lecture 25 NP ClassP = ? NP = ? PSPACEIf P = NP, thenLadner TheoremHow to prove a decision problem belonging to NP?Hamiltonian CycleNondeterministic AlgorithmMinimum Spanning TreeSlide 9How to guess a spanning tree?Co-decision version of MSTAlgorithmco-NPNP ∩ co-NPLinear ProgrammingPrimality TestThereforeProving a problem in NPInteger ProgrammingLecture 26 Polynomial-time many-one reductionA < m BA = Hamiltonian cycle (HC)B = decision version of Traveling Salesman Problem (TSP)HC < m TSPSAT < m 3-SATProperty of < mNP-completeCook Theorem3-SAT is NP-completeHC is NP-completeVertex-Cover is NP-completeProof of Cook TheoremLecture 25 NP ClassP = ? NP = ? PSPACE•They are central problems in computational complexity.If P = NP, then NP-completePLadner Theorem•If NP ≠ P, then there exists a set A lying -between P and NP-complete class, i.e., A is in NP, but not in P and not being NP-compete.How to prove a decision problem belonging to NP?How to design a polynomial-time nondeterministic algorithm?Hamiltonian Cycle•Given a graph G, does G contain a Hamiltonian cycle? •Hamiltonian cycle is a cycle passing every vertex exactly once.Nondeterministic Algorithm•Guess a permutation of all vertices.•Check whether this permutation gives a cycle. If yes, then algorithm halts. What is the running time?Minimum Spanning Tree•Given an edge-weighted graph G, find a spanning tree with minimum total weight. •Decision Version: Given an edge-weighted graph G and a positive integer k, does G contains a spanning tree with total weight < k.Nondeterministic Algorithm•Guess a spanning tree T.•Check whether the total weight of T < k. This is not clear!How to guess a spanning tree?•Guess n-1 edges where n is the number of vertices of G.•Check whether those n-1 edges form a connected spanning subgraph, i.e., there is a path between every pair of vertices.Co-decision version of MST•Given an edge-weighted graph G and a positive integer k, does G contain no spanning tree with total weight < k?Algorithm•Computer a minimum spanning tree. •Check whether its weight > k. If yes, the algorithm halts.co-NP•co-NP = {A | Σ* - A ε NP}NP ∩ co-NP So far, no natural problem has been found in NP ∩ co-NP, but not in P.Linear Programming•Decision version: Given a system of linear inequality, does the system have a solution?•It was first proved in NP ∩ co-NP and later found in P (1979).Primality Test•Given a natural number n, is n a prime? •It was first proved in NP ∩ co-NP and later found in P (2004).Therefore•A natural problem belonging to NP ∩ co-NP is a big sign for the problem belonging to P.Proving a problem in NP•In many cases, it is not hard.•In a few cases, it is not easy.Integer Programming•Decision version: Given A and b, does Ax > b contains an integer solution?•The difficulty is that the domain of “guess” is too large.Lecture 26 Polynomial-time many-one reductionA < m B•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.pA = 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•From a given graph G, we need to construct (n cities, a distance table, k). pSAT < m 3-SAT•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? pProperty of < m•A < m B and B < m C imply A < m C•A < m B and B ε P imply A ε Pppp ppNP-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. pCook TheoremSAT is NP-complete3-SAT is NP-completeHC is NP-completeVertex-Cover is NP-completeProof of Cook


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UTD CS 4384 - Lecture 25- NP Class

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