Following the above rules, construct the dual problem for the following primal problem. PRIMAL Maximize X1 + 2x2 + 3x3 Subjected to X1 - 2x2 + 2x35x4 <= 11 -4x1 + x2 + 2x3 + 2x4 >= 3 2x1 X3 + 3x4 = -1 X1, X2 >= 0 DUAL
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- 4. Reducibility, NP-completeness 4a. Recall that linear ax + b = 0 is reducible to the quadratic ax²+bx+c=0. Show that the quadratic ax2+bx+c=0 is reducible to quartic ax*+bx+cx+dx+e=0 (Lec14.1). Is quartic reducible to quadratic? 4b. Consider the 4-way Partition Problem: Given a set S of positive integers, determine whether S can be partitioned into four subsets S1, S2, S3, and S4 such that the sum of the numbers in S, is equal to S2, S3, and S4. Demonstrate that the problem is NP-complete based on NP-completeness of the 2-way Partition Problem. 4c. Prove that P != NP.8. A school is creating class schedules for its students. The students submit their requested courses and then a program will be designed to find the optimal schedule for all students. The school has determined that finding the absolute best schedule cannot be solved in a reasonablo time. Instead they have decided to use a simpler algorithm that produces a good but non-optimal schedule in a more reasonable amount of time. Which principle does this decision best demonstrate? O A Unreasonable algorithms may sometimes also be undecidable O B. Heuristics can be used to solve some problems for which no reasonable algorithm exsts O C. efficiency Two algorithms that solve the same problem must also have the same O B. Approximate solutions are often identical to optimal solutionsConsider the following recurrence: T(1)=1; T(n) = 2.T()+n, for n> 1, n a power of 3. =..... Find T(27) by substitution, starting with n = 1, n = 65 2 169 29 15 6