1. (25 pt) A palindrome is a string that reads the same from left to right and from right to left. Design an algorithm to find the minimum number of characters required to make a given string to a palindrome if you are allowed to insert characters at any position of the string. For example, for the input “aab” the output should 1 (we’ll add a ’b’ in the beginning so it becomes “baab”).
The algorithm should run in O(n2) time if the input string has length n.
2. (25 pt) Consider the problem of “Approx-3SAT”: The input is the same as 3-SAT, which is a boolean expres-
sion C1^C2^ ^Cn where each Ci is an “or” of three literals (each literal can be one of x1, . . . , xm, :x1, . . . , :xm). Note that we assume a clause cannot contain duplicate literals (e.g., (x1 _ x1 _ x2) is not allowed). Instead of determining whether there’s a truth assignment on fxi gmi=1 that satisfies this boolean expression (which means it satisfies all the clauses), now we want to determine whether there’s an assignment that satisfies at least n 1 clauses. Prove that Approx-3SAT can be polynomial time reduced to 3-SAT.
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