LINEAR PROGRAMMING PROBLEM: Since more than one week I' m not able to solve this problem, the problem does not seem NP , and the "e" stands for the street segment. I would need to know how to write the ILP and to relax it to LP and if is possible the point b) as well.

Exercise 2 n ) 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, Pi 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 Pi,j for each good in such a way that max{Eee Paille E} is as small as possible. (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