We say that $H$ is a matching graph if it contains $2n$ vertices and only $n$ vertex-disjoint edges, i.e. $H$ only contains those $n$ edges and no more.Given a graph $G=(V,E)$ a subset $U\subseteq V$ is called a matching set if the sub graph that is induced by $U$ is a matching graph.Prove that approximating the problem of finding the largest matching set is $NP-Hard$ for any constant factor.I tried reducing to it from the $IS$ problem. I know that it is $NP-Hard$ to approximate $IS$ for any constant factor. I thought of this reduction, given a graph $G=(V,E)$:$1)$ Find $k=\max\limits_{\ v\in V} deg(v)$.$2)$ Construct $k$ graphs exactly like $G$: $G_1(V_1,E_1),\dots,G_k(V_k,E_k)$. those graphs are exactly like $G$ but the vertices are signed by the index and so does the edges. $3)$ Return $H=(\cup_{i} V_i,\cup E_i)$.I wanted to say that:If $m$ is the size of the maximal $IS$ in $G$, the size of the largest matching set in $H$ is: $k\ \cdot m$. I failed doing so. Can someone help?
hardness of constant approximation of largest matching set
cc.complexity theory;approximation algorithms;reductions;approximation hardness