$($Bonus $10pts)$ As seen in class, Min$-$Vertex$-$Cover is NP$-$hard in general graphs. Recall that a vertex cover of a graph $G=(V,E)$ is a set CsubeV that has the property that every edge in $E$ is incident to a vertex in $C.$ Let $M$ be a maximum matching in $G.$

$($a$)$ Describe how to get an approximation for the minimum vertex cover of a graph $G$ from $M.$ What is the approximation factor? Justify.

$($b$)$ We are given a bipartite graph $G=(V,E)$ with $V=RL$ where $L$ and $R$ are the partite sets of vertices. Describe a poly$-$time algorithm that computes a minimum vertex cover of $G$ exactly. Analyze the time complexity. $($Hint: Try a greedy approach starting from $M.$ Be careful to prove that all edges are actually covered.$)$