A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 17.2 Hard Problems 575For the first two cases it is fairly easy to see that the original clauseis satisfiable if and only if the resulting clauses are satisfiable. For thecase were we replaced a clause with more than three literals, consider thefollowing.1. If E is satisfiable, then E ′ is satisfiable: Assume x m is assignedtrue. Then assign z t , t ≤ m − 2 as true and z k , t ≥ m − 1 asfalse. Then all clauses in Case (3) are satisfied.2. If x 1 , x 2 , ..., x j are all false, then z 1 , z 2 , ..., z j−3 are all true. Butthen (x j−1 + x j−2 + z j−3 ) is false.✷Next we define the problem VERTEX COVER for use in further examples.VERTEX COVER:Input: A graph G and an integer k.Output: YES if there is a subset S of the vertices in G of size k or less suchthat every edge of G has at least one of its endpoints in S, and NO otherwise.Example 17.2 In this example, we make use of a simple conversion betweentwo graph problems.Theorem 17.2 VERTEX COVER is N P-complete.Proof: Prove that VERTEX COVER is in N P: Simply guess a subsetof the graph and determine in polynomial time whether that subset is in facta vertex cover of size k or less.Prove that VERTEX COVER is N P-hard: We will assume that K-CLIQUE is already known to be N P-complete. (We will see this proof inthe next example. For now, just accept that it is true.)Given that K-CLIQUE is N P-complete, we need to find a polynomialtimetransformation from the input to K-CLIQUE to the input to VERTEXCOVER, and another polynomial-time transformation from the output forVERTEX COVER to the output for K-CLIQUE. This turns out to be asimple matter, given the following observation. Consider a graph G anda vertex cover S on G. Denote by S ′ the set of vertices in G but not in S.There can be no edge connecting any two vertices in S ′ because, if therewere, then S would not be a vertex cover. Denote by G ′ the inverse graphfor G, that is, the graph formed from the edges not in G. If S is of sizek, then S ′ forms a clique of size n − k in graph G ′ . Thus, we can reduceK-CLIQUE to VERTEX COVER simply by converting graph G to G ′ , and