Wireless Network Design: Optimization Models and Solution ...

10 Integer Programming Models for Power-optimal Trees in Wireless Networks 231
p (ik). But in any optimal integer solution, if there is any flow leaving node i and
arriving at destination t, then the flow amount is one unit, and it is carried by exactly
one of the outgoing arcs of node i. Thus, ∑ N−1
ℓ=k xt (iℓ) ≤ 1, and the same sum is zero if
∑ N−1
ℓ=k y (iℓ) = 0, which proves the validity of (10.8).
In models MET-F1 and MET-F2, a node will use at most one of its candidate
power levels. This corresponds to the following inequalities:
∑
j∈V \{i}
yi j ≤ 1,i ∈ V. (10.9)
Although (10.9) is valid, it is redundant for defining the integer optimum. A
simple proof based on contradiction is to assume that, at integer optimum, there are
multiple y-variables being equal to one at some node. Setting these y-variables other
than the one corresponding to the highest power to zero, it is easily verified that
(10.5) and (10.8) remain satisfied, and the total power decreases. Hence constraints
(10.9) are not needed for the correctness of the models. In fact, they have no impact
on the LP optimum either, as stated in the proposition below.
Proposition 10.2. The LP relaxations of models MET-F1 and MET-F2 have an
optimal solution satisfying (10.9).
Proof. Consider MET-F2. Assume (x,y) satisfies (10.7)-(10.8), and that y ∈ [0,1] |A| .
We can assume that xt is acyclic for all t, which by (10.7) implies that ∑ N−1
ℓ=k xt (iℓ) ≤ 1.
Define ˆy ∈ [0,1] |A| by
�
y(ik), if ∑
ˆy (ik) =
N−1
ℓ=k y (iℓ) ≤ 1,
�
N−1
1 − ∑ℓ=k+1 y � +
(iℓ) , otherwise.
Then ∑ N−1
ℓ=k ˆy (iℓ) = min � 1,∑ N−1
ℓ=k y �
(iℓ) ≤ 1. It follows that (x, ˆy) satisfies (10.8),
and is hence feasible in the LP-relaxation of MET-F2. Since ˆy ≤ y, we also have
∑(i, j)∈A pi j ˆyi j ≤ ∑(i, j)∈A pi jyi j, which completes the proof. For MET-F1 the proof
is similar.
The following proposition states the relation between the LP relaxations of MEP-
F1 and MEP-F2. By this proposition and Proposition 10.1, the LP relaxations of
the models in this section, following the order in which the models are presented,
become increasingly stronger.
Proposition 10.3. LP ∗ (MET-F1) ≤ LP ∗ (MET-F2).
Proof. The inequality is obtained by observing that any feasible multi-commodity
flow in MET-F2 can be transformed into a feasible single-commodity flow in MET-
F1 by defining xi j = ∑t∈D\{s} xt i j ∀(i, j) ∈ A.
In addition to the flow-based models, two cut-based models for MET were proposed
in [10]. Their underlying idea is that for any node set S containing the source