Consider the 2SAT version of the CNF-SAT problem, in which every clause in the given formula...

 

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Consider the 2SAT version of the CNF-SAT problem, in which every clause in the given formula S has exactly two literals. Note that any clause of the form (a V b) can be thought of as two implications, (→ b) and (→ a). Consider a graph G from S, such that each vertex in G is associated with a variable, x, in S, or its negation, . Let there be a directed edge in G from a to b for each clause equivalent to (a b). Show that S is not satisfiable if and only if there is a variable x such that there is a path in G from x to x and a path from x to x. Derive from this rule a polynomial-time algorithm for solving this special case of the CNF-SAT problem. What is the running time of your algorithm? Consider the 2SAT version of the CNF-SAT problem, in which every clause in the given formula S has exactly two literals. Note that any clause of the form (a V b) can be thought of as two implications, (→ b) and (→ a). Consider a graph G from S, such that each vertex in G is associated with a variable, x, in S, or its negation, . Let there be a directed edge in G from a to b for each clause equivalent to (a b). Show that S is not satisfiable if and only if there is a variable x such that there is a path in G from x to x and a path from x to x. Derive from this rule a polynomial-time algorithm for solving this special case of the CNF-SAT problem. What is the running time of your algorithm?