[+] Gap Inapproximability. Gap problem, gapproducing and gappreserving reductions, PCP theorem, max E3X(N)ORSAT, max E3SAT, Label Cover (MaxRep and MinRep), directed Steiner forest, nodeweighted Steiner tree, Unique Games Conjecture.  [Scribe Notes] [src]  
This lecture is about gap problems: distinguishing between good and bad solutions, with a big gap in between those two notions. This perspective is particularly nice because we can turn NPhardness of a decision problem into NPhardness of approximation. It also often leads to tighter bounds on approximability. For example, we'll see a reduction that gives an optimal 7/8inapproximability for Max E3SAT. A big tool here is the PCP (Probabilistically Checkable Proofs) theorem, which won a Gödel Prize in 2001. We'll see how gap problem hardness naturally leads to probabilistically checkable proofs and vice versa. Then we'll see the core PCP lower bound theorems, a problem called label cover (MinRep and MaxRep). Finally, we'll cover the Unique Games Conjecture, which (if true) leads to further improved inapproximability constants. 
Handwritten notes, page 1/7 •
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Handwritten notes, page 1/7 • [previous page] • [next page] • [PDF] 

Slides, page 1/15 •
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http://erikdemaine.org/papers/Tetris_IJCGA/ (slide from Lecture 3) Slides, page 1/15 • [previous page] • [next page] • [PDF] 
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