9 MT263 problems 1. For each of the following s, t-networks, use the given initial flow in the Ford-Fulkerson algorithm to determine a maximum s, t-flow and a minimum s, t-cut. (a) a (7,4) b (8,4) (6,0 (4,4) (2,0) (2,0) s > e f 3,0) 1 t (2,0) (2,0), (4,4) 7 (7,4) d (4,0) c (b) a (18,0) b (14,8) (8,8) (14,4) (10,10) e s (10,2) (8,8) (20,12) f t (18,18) ~(6,6) (16,6) (6,6) (16,12) d c (c) a (7,0) b (8,0) (6,0 (4,0 (2,0) (2,0) s e f (3,0) t (2,0) (2.0) ~ (4,0) (7,0) d (4,0) c 2. The following asks you to prove a lemma from class. Let f be an s, t-flow in an s, t-network D. Let P be an augmenting path and f' : A -> R20 be defined by 8 f(a) + €(P) <f(a) - (P) f'(a) = . f(a) if a is a forward arc of P if a is a backward arc of P otherwise Prove that f' is an s, t-flow, and determine the value of f'. 3. (a) Prove that in an s, t-network in which all the capacities are non-negative integers, there is a maximum flow in which each arc has an integral flow. (b) In the above question you proved that in an s, t-network in which all the capacities are non- negative integers, there is a maximum flow in which each arc has an integral flow. Write down
an application in which finding an integral maximal flow is required. (That is, think of a real- world problem in which you want to find a maximum flow and in which it is vital that you are transporting integral units, rather than, say fractions of a unit.) 4. Let D = (V, A) be an s, t-network with capacities c : A -> R>0 and let f be an s, t-flow on D. Let X CV be such that s E X and te X. Prove that the value of the flow f is equal to 0 1 X X VEX @uEN+(v) f(v, u) - EN-(v) f (u, v) X A 5. Let D = (V, A) be an s, t-network with capacities c : A -> R>0 and let f be an s, t-flow on D. Prove that 0 1 0 1 X X X X VEV QUEN+(v) f (v, u)A = VEV QUEN-(v) f(u, v)^.