# 6.890 Algorithmic Lower Bounds: Fun with Hardness Proofs (Fall '14)

## Prof. Erik Demaine     TAs: Sarah Eisenstat, Jayson Lynch

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 [+] Gap Inapproximability. Gap problem, gap-producing and gap-preserving reductions, PCP theorem, max E3-X(N)OR-SAT, max E3SAT, Label Cover (Max-Rep and Min-Rep), directed Steiner forest, node-weighted Steiner tree, Unique Games Conjecture. 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 NP-hardness of a decision problem into NP-hardness of approximation. It also often leads to tighter bounds on approximability. For example, we'll see a reduction that gives an optimal 7/8-inapproximability 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.
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