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lect19 - 6.841 Advanced Complexity Theory April 15, 2009...

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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 k-coloring 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 k-coloring 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 K-coloring instances to k-coloring 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 K-coloring to 3-coloring. To illustrate the obstacle to showing Lemma 1.3, we'll sketch a linear time reduction for standard K-coloring to standard k-coloring, with k = 3 . We'll convert K-coloring instance G to a 3-coloring 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 . 19-1 19-2 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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This note was uploaded on 04/02/2010 for the course CS 6.841 taught by Professor Madhusudan during the Spring '09 term at MIT.

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lect19 - 6.841 Advanced Complexity Theory April 15, 2009...

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