Lecture 16: NP-complete Problems
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CS 710 : Complexity Theory 1 / 28 / 2010 Lecture 4 : Time - Bounded Nondeterminism
In the previous lecture, we discussed NP-completeness and gave some strong results pertaining to the complexity of SAT, viz. that it is complete for NP under a reduction computable in logarithmic space and polylogarithmic time, and that it is complete for NQLIN (the set of NP problems solvable in quasilinear time) under quasi-linear time mapping reductions. The conclusion that all naturally occ...
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As we’ve already seen in the preceding lecture, two important classes of problems we can consider are P and NP . One very prominent open question is whether or not these two classes of problems are in fact equal. One tool that proves useful when considering this question is the concept of a problem being complete for a class. In the first two portions of this lecture, we show two problems to be...
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As in our previous study of NP, it is useful to identify those problems that capture the essence of PSPACE in that they are the “hardest” problems in that class. We can define a notion of PSPACE-completeness in a manner exactly analogous to NP-completeness: Definition 1 Language L′ is PSPACE-hard if for every L ∈ PSPACE it holds that L ≤p L′. Language L′ is PSPACE-complete if L′ ∈ PSPACE and L′...
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In this lecture, we will begin to talk about the “PCP Theorem” (Probabilistically Checkable Proofs Theorem). Since the discovery of NP-completeness in 1972, researchers had mulled over the issue of whether we can efficiently compute approximate solutions to NP-hard optimization problems. They failed to design such approximate algorithms for most problems. They then tried to show that computing ...
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تاریخ انتشار 2009