Verification of Correctness of the LP Formulation

2.3 Linear Programming Formulation

2.3.2 Verification of Correctness of the LP Formulation

Given this framework, the search for the most efficient download strategy becomes a linear programming (LP) problem, as described in [80] and restated below. In this LP, the optimization variablesfi(e) andx(e) represent the virtual flow to nodeithrough edgee∈ E and the total flow through edge e∈ E, respectively. The virtual flow f_i(e) is the flow over edgeethat is useful to node i. The LP is then given by

minf,x,t d^Tt

s.t. x(e)≥fj(e), ∀j∈ {1, . . . , m}, ∀e∈ E X

v_j^(τ)

x((v^(τ)_i , v_j^(τ)))

c(v_i) ≤tτ, ∀v_i^(τ)∈ V, ∀τ ≤I fj(e)≥0, ∀j ∈ {1, . . . , m}, ∀e∈ E

X

v_k^(τ)

fj((v_k^(τ), v^(τ)_i ))−X

v_k^(τ)

fj((v_i^(τ), v^(τ)_k ))

=











F ifτ =i

−F ifi= 0, τ = 1 0 otherwise











,∀j∈ {1, . . . , m}

(2.2)

The first constraint sets the total flow on each edge to the maximum among all virtual flows over the edge; this value suffices for multicast network coding by [1]. The second constraint requires that the duration of each phase be the maximum, over all nodes, of the time required for nodevto deliver its flow for the given phase; we calculate this value as the total flow out of nodev divided by the upload capacity of v. The third constraint requires that all flows be non-negative. The last constraint guarantees the conservation of flow at all nodes in the network.

In [80], the LP is stated without an explicit proof. We provide a proof in the following section.

We will check the capacity constraint and causal flow constraint (a counterpart of the flow conservation constraint) for these cases.

Now that the constraints in (2.2) are considered at the phase boundaries, the proof for the case where t is at the phase boundary is trivial. Therefore, we focus on proving the correctness of (2.2) for the case where t is within the time slot. For the proof, we first consider necessary and sufficient conditions for a feasible flow at any time instantt. Then, we show that the flow solution obtained from (2.2) satisfies these conditions.

Before starting the proof, we further define some notation.

• Let Ti and ti denote the start time of the i^th time slot and the duration of the time slot, respectively.

• Let Fi(n⁽ⁱ⁾) denote the amount of file at node n⁽ⁱ⁾ at the beginning of the i^th time slot, Ti (whereFi(n⁽ⁱ⁾)≥0, ∀i, ∀n⁽ⁱ⁾ ∈ N⁽ⁱ⁾). In other words, Fi(n⁽ⁱ⁾) is the file size that is transferred from the replica of the node in the previous time slot (n⁽ⁱ⁻¹⁾) over the memory edge, and Fi+1(n⁽ⁱ⁺¹⁾) is the file size that is transferred to the replica of the node in the next time slot (n⁽ⁱ⁺¹⁾) over the memory edge.

• Let h(l) and hp(l) be the total amount of actual flow and the multicast flow for destination pover edge l∈ E_tran⁽ⁱ⁾ in the entire time slot, respectively.

2.3.2.1 Definition of a Feasible Flow

To define a “feasible flow” under our network model, we consider two constraints: (a) the upload capacity constraint of a node and (b) the causal flow constraint of a node. Since there are memory flows in our problem, we consider the causal flow constraint instead of the flow conservation constraint that is used for the static case where there are no memory flows.

Capacity Constraint:

Since we consider a flow problem in the time-expanded graph, the given upload capacity of a node is in terms of a transmission rate, rather than the amount of flow. Therefore, for a capacity constraint, the total instantaneous flow rate out of a node should be no greater than the upload capacity at any time instant. Note that within a time slot, the flow allocation is fixed. We also assume that the flow rate is constant within the time slot.

Then, the instantaneous flow rate over link l ∈ E_tran⁽ⁱ⁾ is h(l)/ti. Now, the upload capacity constraint of a node at any time instantt∈[Ti, Ti+ti) is as follows:

Instantaneous flow rate

z X}| {

l: tail(l) =n⁽ⁱ⁾ l∈ E_tran⁽ⁱ⁾

h(l)/ti ≤

Rate

z }| {

cap(n⁽ⁱ⁾), ∀n⁽ⁱ⁾∈ N⁽ⁱ⁾ (2.3)

Causal Flow Constraint:

In a network without memory, a flow solution is feasible if and only if it satisfies capacity constraints and flow conservation. In contrast, in a network with memory, there are addi- tionally nonnegative memory flows over time, corresponding to received information that is stored at a node. In this case, the flow conservation constraint becomes the causal flow constraint: the total incoming flow up to time instant t is no less than the total outgoing flow up to t (called causality—the difference corresponds to the outgoing memory flow), and each sink node receives entire file by its finish time. Note that the total incoming flow includes the memory flow from the previous phase. Also, recall that for the causal flow constraint as well as the flow conservation constraint, we need to check these constraints for each multicast flow rather than the actual flow.

Now, the causal flow constraint of a node at any time instant t ∈ [Ti, Ti+ti) is as follows: for any peerp,

Total incoming flow in [Ti,t)

z }| {

X

l: head(l) =n⁽ⁱ⁾ l∈ E_tran⁽ⁱ⁾

hp(l)(t−Ti) ti

+

Memory flow from the previous phase

z }| { Fi(n⁽ⁱ⁾)

≥ X

l: tail(l) =n⁽ⁱ⁾ l∈ E_tran⁽ⁱ⁾

hp(l)(t−Ti) ti

| {z }

Total outgoing flow in [Ti,t)

, ∀n⁽ⁱ⁾∈ N⁽ⁱ⁾, ∀p

(2.4)

In addition to the causal flow constraint within the time slot, we need the flow conser-

vation constraint at the phase boundary (t=Ti+ti) as follows: for any peer p, X

l: head(l) =n⁽ⁱ⁾ l∈ E_tran⁽ⁱ⁾

hp(l) +Fi(n⁽ⁱ⁾)− X

l: tail(l) =n⁽ⁱ⁾ l∈ E_tran⁽ⁱ⁾

hp(l)

= Fi+1(n⁽ⁱ⁺¹⁾)

| {z }

Memory flow to the following phase

, ∀n⁽ⁱ⁾∈ N⁽ⁱ⁾, ∀p (2.5)

where for the source node s⁽¹⁾ in the first time slot, F₁(s⁽¹⁾) = F, and for the sink node p⁽ⁱ⁾_i ,F_i+1(p⁽ⁱ⁺¹⁾_i ) =F.

2.3.2.2 Verification of the LP

We show that the flow solution from (2.2) satisfies constraints (2.3)-(2.5) at any time instant twithin the time slot (t∈(Ti, Ti+ti)).

Capacity Constraint: The flow solution x(l) from (2.2) satisfies the capacity con- straint as follows:

X

l: tail(l) =n l∈ E_tran⁽ⁱ⁾

x(l)/cap(n)≤t_i, ∀n∈ N, ∀i

(2.6)

Here, ti is the maximum over all nodes belonging to the i^th time slot. By replacing ti and x(l) with ti and h(l), respectively, we show that (2.6) is equivalent to (2.3), since ti ≥ 0, and cap(n)≥ 0. Therefore, the LP solution is feasible in terms of the capacity constraint at any time instant.

Causal Flow Constraint: The flow conservation constraint in (2.2) is as follows:

X

l:head(l)=n

fp^(k)_k (l)− X

l:tail(l)=n

fp^(k)_k (l)

=











F ifn=p^(k)_k

−F ifn=s⁽¹⁾ 0 otherwise

,∀k∈ {1,2, . . . , m}

(2.7)

Since each summation of incoming and outgoing flows includes the memory flows in

(2.7), we can separate them for clarification as follows:

X

l: head(l) =n⁽ⁱ⁾ l∈ E_tran⁽ⁱ⁾

fp^(k)_k (l) +F_i(n⁽ⁱ⁾)

= X

l: tail(l) =n⁽ⁱ⁾ l∈ E_tran⁽ⁱ⁾

fp^(k)_k (l) +F_i+1(n⁽ⁱ⁺¹⁾)

where







Fi+1(n⁽ⁱ⁺¹⁾) =F ifn⁽ⁱ⁾=p^(k)_k Fi(n⁽ⁱ⁾) =F ifn⁽ⁱ⁾=s⁽¹⁾

(2.8)

Then, (2.8) is equivalent to (2.5) at the phase boundary. Now, let’s consider the causal flow constraint within a time slot (i.e., t∈[Ti, Ti+ti)). The net incoming flows to a node nup to time t is non-negative as shown in (2.9).

X

l: ^{head(l) =n}

l∈ E_tran⁽ⁱ⁾

f_p(k) k

(l)t−Ti

ti

+Fi(n)− X

l: ^{tail(l) =n}

l∈ E_tran⁽ⁱ⁾

f_p(k) k

(l)t−Ti

ti

(a)≥









X

l: ^{head(l) =n}

l∈ E_tran⁽ⁱ⁾

f_p(k) k

(l) +Fi(n)− X

l: ^{tail(l) =n}

l∈ E_tran⁽ⁱ⁾

f_p(k) k

(l)







 t−Ti

ti

(b)= Fi+1(n)t−T_i ti

(c)≥ 0

(2.9)

In (2.9), (a) is from Fi(n) ≥ 0 and 0 ≤ ^t−T_tiⁱ ≤ 1, (b) is from (2.8), and (c) is from non-negativity of F_i+1(n) and ^t−T_t ⁱ

i . Therefore, the flow solution from (2.2) also satisfies (2.4) at any time instanttwithin a time slot. Therefore, the LP solution is feasible in terms of the causal flow constraint at any time instant.

This completes the verification of feasibility.