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HardnessOfVertexCoverandSteinerTree

# HardnessOfVertexCoverandSteinerTree - PCPs and...

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PCPs and Inapproxiability CIS 6930 Sep 17 th , 2009 Hardness of Vertex Cover and Steiner tree problem Lecturer: Dr. My T. Thai Scribe: Nam Nguyen 1 Hardness of Vertex Cover problem In this section, we denote VC(d) the restriction of the cardinality Vertex Cover problem to instances in which each vertex has degree at most d . MAX-3SAT(d) the restriction of MAX-3SAT to Booldean formulae in which each variable occurs at most d times . Theorem 29.13 There is a gap-preserving reduction from MAX-3SAT(29) to VC(30) that trans- forms a Boolean formula φ to a graph G = (V,E) such that If OPT ( φ ) = m , then OPT ( G ) 2 3 | V | , and If OPT ( φ ) < (1 - b ) m , then OPT ( G ) > (1 + v ) 2 3 | V | where v = b 2 Proof: In order to prove this Theorem, we need to construct a transformation between instances of MAX-3SAT(29) and VC(30), or, in other words, we need to trans- form a Boolean formula φ of MAX-3SAT(29) to a graph G = (V,E) of VC(30) in such a way that the two conditions above will follow. Without loss of gen- erality, we can assume that φ has m clauses of exactly 3 literals in CNF form each. Below is the construction of an instance G=(V,E) from φ 1. Let each literal in φ be a vertex in V. 2. For each clause, G has 3 edges connecting its 3 vertices, and For any u, v V, if u and v are negations of each other, there will be an edge connecting u and v. For example, the Boolean formula φ = ( x 1 ¯ x 2 x 3 ) ( ¯ x 1 x 2 x 3 ) has the corresponding graph G as depicted in Figure 1. Besides, using this standard construction, we observe that φ has m clauses of exactly 3 literals each and a literal has a corresponding verex in V. Thus V has exactly 3m vertices ( | V | = 3 m ). Hardness of Vertex Cover and Steiner tree problem-1

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Figure 1 : The graph constructed by φ = ( x 1 ¯ x 2 x 3 ) ( ¯ x 1 x 2 x 3 ) Each vertex in G has 2 edges of type #1 and at most 28 edges of type #2.
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