Consider the following generalization of the maximum flow problem. You are given a directed network G = (V, E) with edge capacities {ce}. Instead of a single (s, t) pair, you are given multiple pairs(s1, t1), ...,(sk, tk), where the si are sources of G and ti are sinks of G. You are also given k demands d1, ..., dk. The goal is to find k flows f (1), ..., f(k) with the following properties: (a)f (i) is a valid flow from st to ti . (b) For each edge e, the total flow f (1) e + f (2) e + ... + f (k) e does not exceed the capacity ce. (c) The size of each flow f (i) is at least the demand di . (d) The size of the total flow (the sum of the flows) is as large as possible. Find a polynomial time algorithm for this problem. Do not give a four part solution for this problem.

Consider the following generalization of the maximum flow problem You are given a directed network G (V, E) with edge capacities [ce]. Instead of a single (s, t) pair, you are given multiple pairs (s1,t1), ..., (Sk, tk), where the s; are sources of G and ti are sinks of G. You are also given k demands d?, ..., dk. The goal is to find k flows f(1),.., f(k) with the following properties: (a) f) is a valid flow from si to ti (b) For each edge e, the total flow f +2 +... + f) does not exceed the capacity ce (c) The size of each flow f(i) is at least the demand di (d) The size of the total flow (the sum of the flows) is as large as possible Consider the following generalization of the maximum flow problem You are given a directed network G (V, E) with edge capacities [ce]. Instead of a single (s, t) pair, you are given multiple pairs (s1,t1), ..., (Sk, tk), where the s; are sources of G and ti are sinks of G. You are also given k demands d?, ..., dk. The goal is to find k flows f(1),.., f(k) with the following properties: (a) f) is a valid flow from si to ti (b) For each edge e, the total flow f +2 +... + f) does not exceed the capacity ce (c) The size of each flow f(i) is at least the demand di (d) The size of the total flow (the sum of the flows) is as large as possible