In task scheduling, it is common to use a graph representation with a node for each task and a directed edge from task $i$ to task $j$ if $i$ is a precondition for $j .$ This directed graph depicts the precedence constraints in the scheduling problem. Clearly, a schedule is possible if and only if the graph is acyclic; if it isn't, we'd like to identify the smallest number of constraints that must be dropped so as to make it acyclic.
Given a directed graph $G=(V, E),$ a subset $E^{\prime} \subseteq E$ is called a feedback arc set if the removal of edges $E^{\prime}$ renders $G$ acyclic.
FEEDBACK ARC SET (FAS): Given a directed graph $G=(V, E)$ and a budget $b,$ find a feedback arc set of $\leq b$ edges, if one exists.
(a) Show that FAS is in NP.
FAS can be shown to be NP-complete by a reduction from VERTEX COVER. Given an instance $(G, b)$ of $\mathrm{VERTEX}$ COVER, where $G$ is an undirected graph and we want a vertex cover of size $\leq b$ we construct a instance $\left(G^{\prime}, b\right)$ of FAS as follows. If $G=(V, E)$ has $n$ vertices $v_{1}, \ldots, v_{n},$ then make $G^{\prime}=\left(V^{\prime}, E^{\prime}\right)$ a directed graph with $2 n$ vertices $w_{1}, w_{1}^{\prime}, \ldots, w_{n}, w_{n}^{\prime},$ and $n+2|E|$ (directed) edges:
$\left(w_{i}, w_{i}^{\prime}\right)$ for all $i=1,2, \dots, n$
$\left(w_{i}^{\prime}, w_{j}\right)$ and $\left(w_{j}^{\prime}, w_{i}\right)$ for every $\left(v_{i}, v_{j}\right) \in E$
(b) Show that if $G$ contains a vertex cover of size $b,$ then $G^{\prime}$ contains a feedback arc set of size $b$
(c) Show that if $G^{\prime}$ contains a feedback arc set of size $b$, then $G$ contains a vertex cover of size (at most) $b$. (Hint: given a feedback arc set of size $b$ in $G^{\prime}$, you may need to first modify it slightly to obtain another one which is of a more convenient form, but is of the same size or smaller. Then, argue that $G$ must contain a vertex cover of the same size as the modified feedback arc set.)