Wireless Network Design: Optimization Models and Solution ...

10 Integer Programming Models for Power-optimal Trees in Wireless Networks 233
10.4 Network Flow Models for RAP
The models in Section 10.3 can be adapted to RAP, by including additional constraints
to ensure strong connectivity between nodes in D by bi-directional links.
For the resulting models, propositions corresponding to those in Section 10.3 apply.
The first flow model for RAP is due to Das et al. [29]. Among the nodes in D,
one is selected as the source node. As with MET, we denote it by s. Utilizing the
underlying idea of MET-Das leads to the following model:
[RAP-Das] min ∑ Pi
i∈V
s. t. (10.1),(10.2),(10.3),(10.4), (10.12)
zi j = z ji,(i, j) ∈ E. (10.13)
In RAP-Das, the last set of constraints (10.13) ensures symmetry. As a result of
the constraints, a link can be used if and only if the power values of both nodes are
high enough to establish bi-directional communication.
A second flow model, obtained by adapting MET-F1 to RAP, is among the models
proposed by Montemanni and Gambardella in [39]. We denote the model by
RAP-MG to indicate the two authors. In [40], RAP-MG has been shown to be
competitive in comparison to RAP-Das [29] and a model using sub-tour elimination
constraints by Althaus et al. [5, 6].
[RAP-MG] min ∑
(i, j)∈A
pi jyi j
s. t. (10.1),(10.5),(10.6),
N−1
xπi(k),i ≤ (|D| − 1) y (iℓ),i ∈ V,k ∈ {1,...,N − 1}.(10.14)
The last set of constraints in RAP-MG plays the same role as (10.13). We remark
that the flow model in [39] is in fact constructed using incremental power variables.
The equivalence to RAP-MG follows from the discussion in Section 10.3.2. In [40],
some valid inequalities for RAP-Das and RAP-MG are presented and used. We do
not discuss these inequalities here, but refer to [40] for details. The inequalities will
however be used for some of the numerical illustrations in Section 10.6.
By introducing symmetry-enforcing constraints to MET-F2, a multi-commodity
flow model for RAP is obtained. We denote it by RAP-F2.
∑
ℓ=k