I have two questions about shortest path

Suppose that we are given n intervals [x_i, y_j] for 1 lessthanorequalto i lessthanorequalto n, which together cover an interval [x_0, y_0]. Each interval [x_i, y_i] has a positive cost c_i. Give an efficient algorithm to a subset of intervals with minimum total cost which together still cover the interval [x_0, y_0]. Consider a digraph D = (V, A) with two distinct vertices s mid t. (a) Let F denote the set of edges in A which appear in some shortest s-t path (in terms of the number of edges) in D. Give an algorithm to output F in O(|V| + |A|) time. (b) Give an efficient algorithm to find an inclusion-wise maximal (not necessarily) edge-disjoint shortest s-t paths in D in O(|V| + |A|) time