The Complexity of Maximum $k$-Order Bounded Component Set Problem

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 $S \subseteq V$ofmaximumsizesuchthateachcomponentintheinducedgraph$G[S]$hasatmost$k$vertices\mathrm{.}Weprovethatforconstant$k$,Max-$k$-OBCS is hard to approximate within a factor of $n^{1-ϵ}$, for any $\epsilon

0$,unless$$\mathsf{P}$ = $\mathsf{N}\mathsf{P}$$. This is an improvement on the previous lower bound of $\sqrt{n}$ for Max-2-OBCS due to Orlovich et al. 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 Tur'an’s greedy algorithm for Max-IS and prove that it approximates Max-$k$-OBCS within a factor of $(2k-1)\bar{d}+k$, where $\bar{d}$ is the average degree of the input graph $G$. This approximation factor is a generalization of Tur'an’s approximation factor for Max-IS.