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complexity1-handouts - ECE 537 Foundations of Computing...

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ECE 537 – Foundations of Computing Module 9, Lecture 1: Complexity Theory – NP-Completeness Professor G.L. Heileman University of New Mexico Fall 2008 c 2008 G. L. Heileman Module 9, Lecture 1
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References Cormen et. al., Chapter 34. M. R. Garey & D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, New York, 1979. C. Papadimitriou, Computational Complexity, Addison Wesley, 1994. Michael Sipser, Introduction to the Theory of Computation, 2nd edition, PWS Publishing Company, 2005. c 2008 G. L. Heileman Module 9, Lecture 1
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Overview Motivation Complexity Classes NP -Completeness Proving a problem NP -Complete c 2008 G. L. Heileman Module 9, Lecture 1
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Motivation There exist many “real-life”, practical problems for which efficient (i.e., polynomial time) algorithms are not known. Furthermore, for many of these problems it has not even been shown if efficient algorithms are possible. We will be considering these problems when we discuss the theory of NP -completeness. It is important for algorithm designers to understand this theory so that they may direct their problem-solving efforts towards approaches that have a possibility of yielding useful algorithms. For example, if we show that a problem is NP -complete, then we probably won’t want to spend much time looking for an efficient algorithm that solves the problem. Many smart people have tried, without success, for many years to find just one polynomial-time solution to an NP -complete problem. c 2008 G. L. Heileman Module 9, Lecture 1
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Motivation So if a problem is NP -complete, do we give up? Not necessarily, we might: Look for algorithms that efficiently solve various special cases of interest (which are not NP -complete). Relax the problem statement so that efficient algorithms can be realized while meeting most of the requirements of the original problem. Look for algorithms that are not guaranteed to run quickly, but on average they do. Rather than seeking an optimal solution, we might consider approximation algorithms that are guaranteed to get close to the optimal solution. c 2008 G. L. Heileman Module 9, Lecture 1
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Motivation Complexity theory involves the investigation of hierarchies of complexity classes. In Module 3 we defined a number of complexity classes, and recall that a complexity class is composed of a set of languages. Remember that complexity hierarchies can be defined w.r.t. any resource consumed by the computation. We will mainly be concerned with the hierarchy based on running time, but there is also one based on space requirements.
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