Data Structures and Algorithm Analysis - Computer Science at ...

Sec. 17.2 Hard Problems 549strategy is to replace any clause C i that does not have exactly three literalswith a set of clauses each having exactly three literals. (Recall that a literalcan be a variable such as x, or the negation of a variable such as x.) LetC i = x 1 + x 2 + ... + x j where x 1 , ..., x j are literals.1. j = 1, so C i = x 1 . Replace C i with C ′ i :(x 1 + y + z) · (x 1 + y + z) · (x 1 + y + z) · (x 1 + y + z)where y and z are variables not appearing in E. Clearly, C i ′ is satisfiableif and only if (x 1 ) is satisfiable, meaning that x 1 is true.2. J = 2, so C i = (x 1 + x 2 ). Replace C i with(x 1 + x 2 + z) · (x 1 + x 2 + z)where z is a new variable not appearing in E. This new pair of clausesis satisfiable if and only if (x 1 + x 2 ) is satisfiable, that is, either x 1 orx 2 must be true.3. j > 3. Replace C i = (x 1 + x 2 + · · · + x j ) with(x 1 + x 2 + z 1 ) · (x 3 + z 1 + z 2 ) · (x 4 + z 2 + z 3 ) · ...·(x j−2 + z j−4 + z j−3 ) · (x j−1 + x j + z j−3 )where z 1 , ..., z j−3 are new variables.After appropriate replacements have been made for each C i , a Booleanexpression results that is an instance of 3 SAT. Each replacement is satisfiableif and only if the original clause is satisfiable. The reduction is clearlypolynomial time.For 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.