Proof of theorem 31

c_u(a₀), min

X⊆V





 X

a∈∆⁻X

c_u(a) − X

a∈∆⁺X\{a0}

c_l(a) + ρ(X) : a₀ ∈ ∆⁺X









. (4.10) If c_l, c_u and ρ are integer valued and (4.10) is finite, then there exists an integer–valued maximum flowφ : A → Z.

2. Proof of theorem 31

In this section we first construct an auxiliary graph G(V, A) for the LDRN N . Then we define a submodular flow problem on the graph G which is equivalent to finding a flow in our LDRN. The proof of Theorem 31 will follow from an application of Theorem 32 to this submodular flow problem on graph G.

The auxiliary graph: Define G = (V, A) by

V = {ν_i(j) : 2 ≤ i ≤ M − 1, 1 ≤ j ≤ m_i} ∪

{υi(j) : 2 ≤ i ≤ M − 1, 1 ≤ j ≤ mi} ∪ {υ1(1), νM(1)}

and

A = {a_i(j) : 2 ≤ i ≤ M − 1, 1 ≤ j ≤ m_i} ∪ {a₀}

where a_i(j) = (ν_i(j), υ_i(j)) and a₀ = (ν_M(1), υ₁(1)). (See Figure 19 for an example of graph G.)

The submodular flow problem on graph G: Fix a positive integer R₀. Let c_u(a) = +∞

for a ∈ A\ {a₀}, c_u(a₀) = R₀, and c_l(a) = 0 for every edge a ∈ A. To define the function

a

Fig. 19. Graph G corresponding to the LDRN in Figure 18.

ρ : 2^V → R ∪ {+∞} we need some definitions. For any H ⊆ V and 1 ≤ i ≤ M − 1 let

Ji(H) , {j : υⁱ(j) /∈ H} , K_i(H) , {k : νi+1(k) ∈ H} .

Consider the correspondence of the indices of the node set V and the node set V and define

ρi(H) , rank(Gⁱ[Ki(H), Ji(H)])

Lemma 33 The function ρ defined above is submodular and satisfies ρ(∅) = ρ(V) = 0.

Proof To establish the submodularity of ρ, we need to show that for every H₁, H₂ ⊆ V,

ρ(H₁) + ρ(H₂) ≥ ρ(H₁∩ H₂) + ρ(H₁ ∪ H₂).

Fix 1 ≤ i ≤ M − 1, and for H ⊆ V let

F_i(H) = {j : υ_i(j) ∈ H} .

Notice that F_i(H₁∩ H₂) = F_i(H₁) ∩ F_i(H₂), K_i(H₁∩ H₂) = K_i(H₁) ∩ K_i(H₂), F_i(H₁∪ H₂) = F_i(H₁) ∪ F_i(H₂), and K_i(H₁ ∪ H₂) = K_i(H₁) ∪ K_i(H₂). Let Q_i = {1, · · · , m_i} . Then

ρ_i(H) = rank(G_i[K_i(H), Q_i\F_i(H)]), and

ρ_i(H₁) + ρ_i(H₂) = rank(G_i[K_i(H₁), Q_i\F_i(H₁)]) + rank(G_i[K_i(H₂), Q_i\F_i(H₂)])

(a)

≥ rank(G_i[K_i(H₁) ∩ K_i(H₂), Q_i\ (F_i(H₁) ∩ F_i(H₂))]) + rank(G_i[K_i(H₁) ∪ K_i(H₂), Q_i\ (F_i(H₁) ∪ F_i(H₂))])

= ρ_i(H₁∩ H₂) + ρ_i(H₁∪ H₂),

where (a) follows by Lemma 28. From the preceding inequality and the definition of ρ it follows that

ρ(H₁) + ρ(H₂) =

M −1

X

i=1

(ρ_i(H₁) + ρ_i(H₂))

≥

M −1

X

i=1

(ρ_i(H₁∩ H₂) + ρ_i(H₁∪ H₂))

= ρ(H₁∩ H₂) + ρ(H₁∪ H₂),

which is the desired inequality.

Next observe that Ki(∅) = ∅ and J_i(V) = ∅ for every 1 ≤ i ≤ M − 1. Thus ρ_i(∅) = ρ_i(V) = 0 for every 1 ≤ i ≤ M − 1. Therefore ρ(∅) = ρ(V) = 0.

 The functions c_l, c_u, and ρ defined above are the setting for a submodular flow problem.

The submodular flow problem is feasible because the flow φ(a) = 0 for every a ∈ A is always a feasible flow.

Let Φ be the set of all integer functions φ : A → Z with φ(a0) ≤ R₀. Also let D be the set of all integer vectors d = (`<sub>i</sub>(j) : 1 ≤ i ≤ M, 1 ≤ j ≤ m<sub>i</sub>) with `₁(1) = `<sub>M</sub>(1) ≤ R<sub>0</sub>. Define the bijection T : Φ → D by T (φ) = (`_i(j) : 1 ≤ i ≤ M, 1 ≤ j ≤ m_i) , where

`_i(j) = φ (a_i(j)) , 2 ≤ i ≤ M − 1, 1 ≤ j ≤ m_i

`<sub>1</sub>(1) = `_M(1) = φ(a₀).

Lemma 34 The flow φ ∈ Φ is a feasible submodular flow for graph G if and only if the vectord = T (φ) is a feasible flow in the corresponding network N .

Notice that since for any feasible flow d = (`i(j) : 1 ≤ i ≤ M, 1 ≤ j ≤ mi) in network N with rate R we have `1(1) = `<sub>M</sub>(1) = R, the set D includes all feasible flow vectors for N . Furthermore, the rate R for a feasible flow d = (`_i(j) : 1 ≤ i ≤ M, 1 ≤ j ≤ m_i) is the value of φ(a₀), where φ = T⁻¹(d). Next we prove Lemma 34.

Proof First suppose that φ ∈ Φ is a feasible submodular flow. We begin by showing that d = T (φ) is a feasible flow in N . Let d = (`_i(j) : 1 ≤ i ≤ M, 1 ≤ j ≤ m_i). First notice that d is an integer vector with non–negative elements since 0 = c_l(a) ≤ φ(a). Next we show that for every 2 ≤ i ≤ M − 1,

mi

X

j=1

`<sub>i</sub>(j) = `₁(1) = `_M(1).

Fix a value of 1 ≤ i ≤ M −1 and let H = {υ_i(1), · · · , υ_i(m_i)} ∪ {ν_i+1(1), · · · , ν_i+1(m_i+1)}.

Then K_j(H) = ∅ for j 6= i and J_i(H) = ∅. This implies that for every 1 ≤ j ≤ M − 1, ρ_j(H) = 0, and hence ρ(H) = 0. Now by the second condition of a feasible submodular flow and the fact that ρ(H) = 0 we have

X

a∈∆⁺H

φ(a) ≤ X

a∈∆⁻H

φ(a).

For i = 1 the above condition is

m2

X

j=1

φ (a₂(j)) ≤ φ(a₀). (4.11)

For 2 ≤ i ≤ M − 2 the condition is

mi+1

X

j=1

φ (a_i+1(j)) ≤

mi

X

k=1

φ (a_i(k)) . (4.12)

For i = M − 1 the condition is

φ(a₀) ≤

mM −1

X

j=1

φ (a_{M −1}(j)) . (4.13)

Inequalities (4.11), (4.12), (4.13) imply that

mi

X

j=1

φ(a_i(j)) = φ(a₀)

for 2 ≤ i ≤ M − 2, which proves our claim.

Let R = φ(a0). Next we show that

d_i = (`<sub>i</sub>(1), · · · , `_i(m_i); `<sub>i+1</sub>(1), · · · , `_i+1(m_i+1))

is a rate–R flow for matrix G_ifor 1 ≤ i ≤ M −1. Fix 1 ≤ i ≤ M −1, K ⊆ {1, · · · , m_i+1} , and J ⊆ {1, · · · , m_i} . Let H = {υ_i(j) : j /∈ J} ∪ {ν_i+1(k) : k ∈ K} . Then for every l 6= i, we have K_l(H) = ∅ and hence ρ_l(H) = 0. Also, J_i(H) = J and K_i(H) = K, and hence ρi(H) = rank(Gi[K, J ]). Therefore ρ(H) = ρi(H) = rank(Gi[K, J ]). The second condition of a feasible submodular flow implies that

X

a∈∆⁺H

φ(a) − X

a∈∆⁻H

φ(a) ≤ rank(G_i[K, J ]).

We have ∆⁺H = {a_i+1(k) : k ∈ K} and ∆⁻H = {a_i(j) : j /∈ J} . ThusP and d is a feasible flow for network N .

Our next step is to show that if d = (`i(j) : 1 ≤ i ≤ M, 1 ≤ j ≤ mi) is a feasible rate–R flow in network N and d ∈ D, then φ = T<sup>−1</sup>(d) is a feasible submodular flow in G. Notice that the first property of a feasible submodular flow is satisfied because d is a non–negative vector and hence for any a<sub>i</sub>(j) ∈ A\ {a<sub>0</sub>} , φ(a<sub>i</sub>(j)) = `_i(j) is a non–

negative integer. Furthermore, φ(a₀) = `₁(1) and 0 ≤ φ(a₀) ≤ R₀. Next fix H ⊆ V. For the second property we have to show that:

X

To arrive at (4.14), observe that each a ∈ ∆⁺H corresponds to exactly one i such that a ∈ ∆⁺H_i and each a ∈ ∆⁻H corresponds to exactly one i such that a ∈ ∆⁻H_i. Next suppose that for some i, a ∈ ∆⁺H_ibut a /∈ ∆⁺H. This implies that there exists exactly one j such that a ∈ ∆⁻Hj. Therefore the terms φ(a) and −φ(a) cancel each other out on the right hand side. Similarly if for some i, a ∈ ∆⁻H_i but a /∈ ∆⁻H, then there exists exactly one j such that a ∈ ∆⁺H_j. Therefore the terms −φ(a) and +φ(a) cancel each other out on

the right–hand side. Therefore the right–hand side of (4.14) is equal to its left–hand side. feasibility of di we have

X

Summing the preceding inequalities for 1 ≤ i ≤ M − 1 and by using (4.14) and (4.15) we find that φ satisfies the second property of a submodular flow.

 Next we prove the following lemma:

Lemma 35 The maximum feasible submodular flow φ(a₀) in graph G is min {R₀, C} . Proof We use Theorem 32 to prove this result. We need to find the value of

γ = min

Since c_u(a₀) = R₀ and c_l(a) = 0 for every a ∈ A, we have

γ = min

"

R₀, min

X⊆V

( X

a∈∆⁻X

c_u(a) + ρ(X) : a₀ ∈ ∆⁺X )#

.

Furthermore, since for every a ∈ A\ {a₀}, c_u(a) = +∞, we can restrict the minimization to the subsets X such that ∆⁻X = ∅. Therefore if for 2 ≤ i ≤ M − 1, ν_i(j) /∈ X, we should also have υi(j) /∈ X. In addition, since a0 ∈ ∆⁺X, we have νM(1) ∈ X and υ₁(1) /∈ X. Next suppose that for X ⊆ V we have ∆⁻X = ∅, a₀ ∈ ∆⁺X, and for some 2 ≤ i ≤ M − 1 and 1 ≤ j ≤ m_iwe have νi(j) ∈ X and υ_i(j) /∈ X. Set X⁰ = X ∪ {υ_i(j)} . We still have ∆⁻X⁰ = ∅ and a₀ ∈ ∆⁺X⁰. Also for every l 6= i, Kl(X) = K_l(X⁰) and J_l(X⁰) = J_l(X). Thus, ρ_l(X⁰) = ρ_l(X). But we have J_i(X⁰) = J_i(X) \ {j} and K_i(X⁰) = K_i(X). Therefore, rank(G_i[K_i(X⁰), J_i(X⁰)]) ≤ rank(G_i[K_i(X), J_i(X)]) and ρ_i(X⁰) ≤ ρ_i(X). Consequently ρ(X⁰) ≤ ρ(X). This shows that for computing γ, X can be ignored given the set X⁰. In this way we can consider the minimization only over the sets X ⊆ V such that a0 ∈ ∆⁺X and for every 2 ≤ i ≤ M − 1 and 1 ≤ j ≤ mi, either νi(j) and υi(j) are both in X or are both out of X. Call the set of all such sets X and let M be the set of all cuts in network N that separate s from t. Consider the bijection B : X → M in which B(X) = {s}S {v_i(j) : 2 ≤ i ≤ M − 1, 1 ≤ j ≤ m_i, υ_i(j) /∈ X} . In words, if for 2 ≤ i ≤ M − 1 and 1 ≤ j ≤ m_i, the nodes ν_i(j) and υ_i(j) are both out of X we include v_i(j) in B(X) and if are both in X we exclude v_i(j) from X. Furthermore, we let s ∈ B(X) and t /∈ B(X). In this way every separating cut in network N corresponds to a set X ∈ X and vice versa. Observe that Gi[Ki(X), Ji(X)] is the transfer function from the nodes υi(j) /∈ X to the nodes ν_i+1(k) ∈ X in graph G, which is the same as the transfer function from the nodes vi(j) ∈ B(X) to the nodes v_i+1(k) /∈ B(X) in N . Therefore

G_i[K_i(X), J_i(X)] = G_i[B(X)]. Thus

Therefore for every X ∈ X, ρ(X) = C(B(X)), and the result follows by

γ = min Therefore the second part of Theorem 32 implies that a submodular flow with φ(a₀) = min {R₀, C} is feasible in G with integer values for every integer R₀ ≥ 0. By Lemma 34, all flows with rate R0 ≤ C are achievable in N , proving Theorem 31.