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Unformatted text preview: 6.841 Advanced Complexity Theory April 15, 2009 Lecture 19 Lecturer: Madhu Sudan Scribe: Alex Arkhipov 1. Review of Last Class Last class we gave a formulation of Probabilistically Checkable Proofs as a col oring of a graph that satis es certain constraints. De nition 1.1. The graph kcoloring problem is as follows: Given a graph G = ( E,V ) , does there exist a coloring χ : V → { 1 , 2 ,...,k } such that for each ( u,v ) ∈ E , χ ( u ) 6 = χ ( v ) ? In generalized graph coloring, each edge restricts the coloring of its endpoints by an arbitary relation, described by an admissibility function Π . De nition 1.2. The generalized kcoloring problem is as follows: Given a graph G = ( E,V, Π) , which includes map Π : E ×{ 1 , 2 ,...,k }× { 1 , 2 ,...,k } → { , 1 } , does there exist a coloring χ : V → { 1 , 2 ,...,k } such that for each e = ( u,v ) ∈ E , Π( e,χ ( u ) ,χ ( v )) = 1 ? In a coloring χ , we say an edge e = ( u,v ) is invalid if it does not satisfy the constraint Π( e,χ ( u ) ,χ ( v )) = 1 . The unsatis ability UNSAT ( G,χ ) is fraction of invalid edges in G , and the unsatis ability UNSAT ( G ) of a graph is the minimum UNSAT ( G,χ ) over all colorings χ . Recall the Lemma that we wished to prove that would allow us to reduce the number of colors: Lemma 1.3. There exists a k and δ > , so that for any K , there is a reduction function f from Kcoloring instances to kcoloring instances so that for any G and ˜ G = f ( G ) , • If UNSAT ( G ) = 0 , then UNSAT ( ˜ G ) = 0 . • UNSAT ( ˜ G ) ≥ δ UNSAT ( G ) We'll rst see look at a naive attempt to perform this reduction, and see how it can lead to unsatis ability falling by more than a constant factor δ . 1.1. Attempt at reduction from Kcoloring to 3coloring. To illustrate the obstacle to showing Lemma 1.3, we'll sketch a linear time reduction for standard Kcoloring to standard kcoloring, with k = 3 . We'll convert Kcoloring instance G to a 3coloring instance ˜ G by replacing each edge of G with a gadget of ˜ G that encodes the same restriction. However, we'll nd that if UNSAT ( G ) = ε , then UNSAT ˜ G ≤ ε K , and thus cannot satisfy UNSAT ( ˜ G ) ≥ δ UNSAT ( G ) for any constant δ . 191 192 Figure 1.1 . The conversion of a vertex of G to one of ˜ G . The vertices are marked with their possible colors. 1.1.1. Construction of ˜ G . Make three special nodes { r,g,b } and connect them with edges, so that they must be di erent colors which we'll label red, green, and blue, which we will also call the three possible colors of the nodes. We may restrict the possible colors of a node in ˜ G by connecting it to each of { r,g,b } we want to exclude....
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 Spring '09
 MadhuSudan
 Quadratic equation, satisfying assignment

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