When Bobby’s friends visit her on the farm, she likes to show them around. The farm consists of n fields numbered 1, 2, . . . , n, the first of which contains her house and the n-th of which contains the big barn. A total of m directed paths connect the fields. The i-th path is from field ui to field vi, and has a nonzero length wi. 1 Suppose there are k friends coming today. To show off the farm in the best way, Bobby designs a tour for each friend that starts at her house, potentially travels through some fields, and ends at the barn. Bobby wants the total length of the k tours to be as short as possible, where the length of a tour is the total length of paths in the tour. However, she doesn’t want any two friends to share any path. That is, for any two friends, their tours should not have any common paths (but they can travel through the same fields). Write an LP that can calculate the shortest total length of k tours that satisfies Bobby. You can assume that at least one feasible solution exists. (If you use min-cost flow as an intermediate step, you will still need to provide an LP formulation in the end. Please avoid using shorthand notations such as out-in notation.)

You are organizing a programming competition, where contestants implement Dijkstra's algorithm. Given adirected graph G = (V, E) with integer-weight edges and a starting vertex s ∈ V , their programs are supposedto output triplets (v, v.d, v.π) for each vertex v ∈ V . Design an O(V +E) time algorithm that takes as inputthe original graph G in both adjacency matrix (G.M) and adjacency list (G.Adj) representations, startingvertex s, and the output of a contestant's program (given as an array A of triplets), and returns whetherA is the correct output for G. Write down the pseudocode for your algorithm, explain why it correctlyveries the output, and analyze your algorithm's running time. You may assume that all edge weights of the input graph provided to the contestantsare nonnegative and A (the output of their programs) is in the valid format, i.e., you don't need to verifythat A is actually an array of triplets, with v and v.π being valid vertices and v.d being an integer.Can you…