The Complexity of Maximum k-Order Bounded Component Set Proble

Given a graph G = (V, E) and a positive integer k, in MAXIMUM k-ORDER BOUNDED COMPONENT SET (MAX-k-OBCS), 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, MAX-k-OBCS is hard to approximate within a factor of n1−e, for any e > 0, unless P = NP. This is an improvement on the previous lower bound of √ n for MAX-2-OBCS due to Orlovich et. el. [1]. We provide lower bounds on the approximability when k is not a constant as well. MAX-k-OBCS can be seen as a generalization of MAXIMUM INDEPENDENT SET (MAX-IS). We generalize Turán’s greedy algorithm for MAX-IS and prove that it approximates MAX-k-OBCS 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.