Wireless Network Design: Optimization Models and Solution ...

230 Dag Haugland and Di Yuan
As stated by constraint (10.5), flow can be assigned to arc (i,πi(k)) only if at least
one of y (ik),...,y (i,N−1) equals one, i.e., only if the power of node i equals at least
p (ik). Constraints (10.1) ensure again that there is a path of such arcs from the source
to each destination. In comparison to MET-Das, the numbers of variables and constraints
have gone down by |V | and |A|, respectively, but the coefficient matrix is
denser.
Proposition 10.1. LP∗ (MET-Das) ≤ LP∗ (MET-F1).
Proof. Assume (x,y) is a feasible solution to LP-MET-F1, and define z (ik) =
min � ∑ N−1
ℓ=k y (iℓ),1 � and Pi = max j:(i, j)∈A pi jzi j. If G has a directed cycle such that
x is positive on each arc in the cycle, we can, without violating the constraints or
altering the objective function value of LP-MET-F1, send flow in the reverse direction
until at least one arc gets zero flow and the flow on all other arcs remains
non-negative. We can thus assume that the flow vector x is acyclic, which means
that (10.1) implies xi j ≤ |D| − 1 ∀(i, j) ∈ A. Hence (10.2) is satisfied for all i,k
for which z (ik) = 1, and if z (ik) = ∑ N−1
ℓ=k y (iℓ), (10.2) follows from (10.5). Therefore,
(x,y,P) is feasible in LP-MET-Das, and by the definition of P and z, we get ∑i∈V Pi
= ∑i∈V max j∈V :(i, j)∈A pi jzi j
≤ ∑i∈V maxk=1,...,N−1 p (ik) ∑ N−1
ℓ=k y (iℓ) ≤ ∑i∈V maxk=1,...,N−1 ∑ N−1
ℓ=k p (iℓ)y (iℓ)
= ∑i∈V ∑ N−1
ℓ=1 p (iℓ)y (iℓ) = ∑(i, j)∈A pi jyi j, which completes the proof.
Introducing multi-commodity flow variables, where each commodity corresponds to
a unique destination and vice versa, often leads to stronger formulations for network
design problems. We now go on to demonstrate how this can be accomplished in the
case of MET.
For consistency, we will reuse the variable notation x to denote flows in a multicommodity
model of MET. Doing so will not cause ambiguity, as the precise meaning
of x will be clear from the context.
x t i j
= Flow to destination t ∈ D \ {s} on arc (i, j).
In compact notation, we have defined x ∈ ℜ |A|(|D|−1)
+ . For each destination t, we
denote its components in x, i.e., the flow vector of t, by xt . The multi-commodity
flow model for MET is as follows:
[MET-F2] min ∑
(i, j)∈A
pi jyi j
s. t. (10.6),
x t ∈ F (G,bst),t ∈ D \ {s}, (10.7)
N−1
∑ x
ℓ=k
t (iℓ) ≤
N−1
∑ y (iℓ),i ∈ V,k ∈ {1,...,N − 1},t ∈ D \ {s}.(10.8)
ℓ=k
Constraint (10.8) is a strengthened version of the more intuitive constraint x (ik) ≤
∑ N−1
ℓ=k y (iℓ), stating that arc (i,πi(k)) can carry flow only if the power at i is at least