Wireless Network Design: Optimization Models and Solution ...

234 Dag Haugland and Di Yuan
[RAP-F2] min ∑
(i, j)∈A
pi jyi j
s. t. (10.6),(10.7),(10.8),
N−1
∑ x
ℓ=k
t πi(ℓ),i ≤
N−1
∑ y (iℓ),i ∈ V,k ∈ {1,...,N − 1},t ∈ D \ {s}.
ℓ=k
(10.15)
As with the models in the previous section, constraints (10.9) are valid but redundant
for defining the integer optimum of RAP, as well as at LP optimum. We
formulate the latter by a proposition, and omit the proof as it is very similar to the
one for Proposition 10.2.
Proposition 10.4. The LP relaxations of models RAP-MG and RAP-F2 have an
optimal solution satisfying (10.9).
In terms of LP strength, we have observations similar to those in Section 10.3,
leading to the following proposition. We omit the proofs as they can be easily obtained
by adapting those of Propositions 10.1 and 10.3.
Proposition 10.5. LP ∗ (RAP-Das) ≤ LP ∗ (RAP-MG) ≤ LP ∗ (RAP-F2).
10.5 A Strong Multi-tree Model for RAP
Khoury, Pardalos and Hearn [34] have formulated the minimum Steiner tree problem
by exploiting the idea that any Steiner tree can be considered as |D| Steiner
arborescences rooted at distinct destinations. One can use a binary variable to indicate
whether or not an arc in A is in the arborescence rooted at t ∈ D. This leads to
the multi-tree formulation for the problem, the strength of which has been analyzed
in depth by Polzin and Daneshmand [45].
In this section, we apply the multi-tree idea, and introduce the variables
qt i j =
�
1 if arc (i, j) is used by node i to reach node t ∈ D in the tree
0 otherwise.
10.5.1 The Model
In addition to the q- and y-variables, the latter being defined in Section 10.3.1, the
multi-tree model uses two more sets of binary variables. Note the reuse of notation
z. The reason is that, except for being defined for edges instead of arcs, its meaning
is identical to the z-variables in the previous sections. Let