The Complexity of Maximum k-Order Bounded Component Set Proble
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Given a graph G=(V, E) and a positive integer k, in Maximum k-Order Bounded Component Set problem (k), it is required to find a vertex set S ⊆ V of maximum size such that each component in the induced graph G[S] has at most k vertices. We prove that for constant k, k is hard to approximate within a factor of n^1 -ϵ, for any ϵ > 0, unless P = NP. We provide lower bounds on the approximability when k is not a constant as well. k can be seen as a generalization of Maximum Independent Set problem (Max-IS). We generalize Turán's greedy algorithm for Max-IS and prove that it approximates k within a factor of (2k - 1)d + k, where d is the average degree of the input graph G. This approximation factor is a generalization of Turán's approximation factor for Max-IS.
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