assignment6 - CSIS 0250B Design and Analysis of Algorithms...

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Unformatted text preview: CSIS 0250B Design and Analysis of Algorithms Assignment 6 Due: 11:55 PM, 1 May, 2009 1. Let G = ( V,E ) be a degree- d , undirected graph, where d > 2. (I.e., every node has at most d neighbors.) An unrelated vertex set U of G is a subset of V such that no two vertices u,v ∈ U are connected by an edge in G . The problem of finding the maximum unrelated vertex set is NP-hard. Give a polynomial-time algorithm that guarantees to find an unrelated vertex set with at least n/ ( d + 1) vertices, where | V | = n . Justify your answer. Note that a maximum unrelated vertex set has at most n nodes. The above algorithm yield an approximation ratio of 1 / ( d + 1). Suppose that G satisfies an extra property that one of its vertexes has degree strictly less than d . Give a polynomial-time algorithm that guarantees to find an unrelated vertex set with at least n/d . 2. Again, consider a degree- d , undirected graph G = ( V,E ). A matching M of G is a subset of E such that no two edges in M have a common endpoint.have a common endpoint....
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