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**Unformatted text preview: **10/19/2009 CSE 5311 FALL 2009 KUMAR 1 NP-Complete Problems P and NP Polynomial time reductions Satisfiability Problem, Clique Problem, Vertex Cover, and Dominating Set 10/19/2009 CSE 5311 FALL 2009 KUMAR 2 Polynomial Algorithms Problems encountered so far are polynomial time algorithms The worst-case running time of most of these algorithms is O(n k ) time, for some constant k. All problems cannot be solved in polynomial time There are problems that cannot be solved at all Unsolvable There are problems that can be solved but not in O(n k ) time for some constant k. Note that most useful algorithms of complexity O(n 3 ) or better. Problems that are solvable in polynomial time by polynomial- time algorithms are said to be tractable (or easy or efficient ). Problems that require superpolynomial time are said to be intractable or hard . 10/19/2009 CSE 5311 FALL 2009 KUMAR 3 P and NP Class P problems are solvable in polynomial time. Class NP problems are verifiable in polynomial time. Class NP is the class of decision problems that can be solved by non-deterministic polynomial algorithms . For example, given a problem, we can verify the solution in polynomial time Any problem in P is also in NP P NP It is NOT KNOWN whether P is a proper subset of NP. An NP-complete problem is in NP As difficult as any other problem in NP Can be reduced to an instance of know NP-Complete problem in polynomial time 10/19/2009 CSE 5311 FALL 2009 KUMAR 4 NP What are NP-complete Problems ? A problem is said to be NP-Complete if it is as hard as any problem in NP. No polynomial time algorithm has yet been discovered for the NP-Complete problem. However, it has not been proven that NO polynomial time algorithm can exist for an NP-Complete problem. This problem was first posed by Cook in 1971. The issue of, P=NP or P NP is an open research problem In 2002 a known NP-Complete problem was shown to have a polynomial time algorithm. Whether a given integer is a prime or composite 10/19/2009 CSE 5311 FALL 2009 KUMAR 5 Examples of NP-Complete problems Shortest vs. longest simple paths Finding the shortest paths from a single source in a directed graph G =(V,E) can be completed in O(V + E) time. Even with negative edge weights. However, finding the longest simple path between two vertices is NP- complete. It is NP-Complete even if each of edge weights is equal to one. An Euler tour of a connected directed graph G=(V,E), can be completed in O(E) time. However, the Hamiltonian Cycle is a NP-Complete problem....

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