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Exercise 2 You are tasked with shipping a number n of goods gi {1,..., n} to a target location t. In your very rural area, the roads are in bad shape and often blocked by trees and the like, and also, there are only three cargo companies you can use for the transport of any good gi. You have to order one truck for each good, and know the companies' routes Pi,1, P1,2 and Pi,3 they would take to all specific locations, where each route consists of a sequence of road segments that it uses. Now since you are aware of the frequent road blockages, and you want not too many of your transports to be obstructed, your aim is to pick the paths/companies for the goods in such a way that a single blocked road can intercept as few shipments as possible. I.e. choose a path Pij for every good gi such that maxeee {l{Pijle e P.}|} is as small as possible, where E is the set of all road segments. (a) Formulate the problem as an ILP, and relax to an according LP. (b) Give a rounding algorithm to compute a feasible solution starting from the LP optimum that guarantees a 3- approximation to the optimal solution, and prove its approximation guarantee