UTD CS 4384 - Lecture 25- NP Class (32 pages)

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Lecture 25- NP Class



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Lecture 25- NP Class

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Lecture Notes


Pages:
32
School:
The University of Texas at Dallas
Course:
Cs 4384 - Automata Theory

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Lecture 25 NP Class P NP PSPACE They are central problems in computational complexity If P NP then P NP complete Ladner 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 NPcompete 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 A 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 coNP 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



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