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Let s multiply out the first equation to get it into polynomial standard form When we do that, the system becomes y = x 3 + 3x 2 + 3x + 1 and y = x 3 + 2x 2 + x We can mix the right sides of these equations, getting x 3 + 3x 2 + 3x + 1 = x 3 + 2x 2 + x

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Now let s subtract the entire right side of this equation from the left side When we do that, we obtain x 2 + 2x + 1 = 0 which factors into (x + 1)2 = 0 This equation has the single real root x = 1, with multiplicity 2 When we plug that into the first original equation, we get y = (x + 1)3 = ( 1 + 1)3 = 03 =0 Therefore, the original system has the single real solution ( 1,0), with multiplicity 2

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81 Optimization versus search Recall the traveling salesman problem: Input: A matrix of distances; a budget b Output: A tour which passes through all the cities and has length b, if such a tour exists The optimization version of this problem asks directly for the shortest tour

Question 27-10

Consider the following pair of equations: y = (x + 1)3 and y = (x + 2)3 How can we find the real solutions of this two-by-two system, if any exist

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TSP - OPT Input: A matrix of distances Output: The shortest tour which passes through all the cities TSP

Answer 27-10

Let s multiply both of these equations out to get cubics in polynomial standard form That gives us y = x 3 + 3x 2 + 3x + 1 and y = x 3 + 6x 2 + 12x + 8 When we mix the right sides of these equations, we get x 3 + 3x 2 + 3x + 1 = x 3 + 6x 2 + 12x + 8 Subtracting the entire left side from the right side and then switching right-to-left, we obtain 3x 2 + 9x + 7 = 0

82 Search versus decision Suppose you have a procedure which runs in polynomial time and tells you whether or not a graph has a Rudrata path Show that you can use it to develop a polynomialtime algorithm for R UDRATA PATH (which returns the actual path, if it exists) 83 S TINGY SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, nd a satisfying assignment in which at most k variables are true, if such an assignment exists Prove that STINGY SAT is NP-complete 84 Consider the CLIQUE problem restricted to graphs in which every vertex has degree at most 3 Call this problem CLIQUE -3 (a) Prove that

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