Wireless Network Design: Optimization Models and Solution ...

10 Integer Programming Models for Power-optimal Trees in Wireless Networks 243
performance of RAP-MG+. First, the valid inequalities turned out to speed up B&B
significantly, in particular in obtaining high-quality integer solutions. Second, the
solver (CPLEX) is able to utilize the structure of RAP-MG+ and strengthen further
the LP by its built-in cut generator.
Model RAP-F2 has a performance comparable to that of RAP-MG+, although
the former admits slightly more instances to be solved to optimality. The B&B nodes
used is clearly fewer for RAP-F2. This, per se, does not necessarily mean an advantage.
One can indeed observe that the final gap values as well as the times for
getting integer solutions are in the same range for these two models. Note that the
time values in Table 10.4 do not include those for solving LPs. Thus the overall time
requirement of RAP-F2 is higher.
RAP-MT is the only model that could be solved to integer optimality for all
the instances within the time limit. The extremely sharp LP bound of RAP-MT
virtually eliminates the need of B&B enumeration – Integer optimum is found and
verified without B&B for all but one instance, which required two B&B nodes.
As for solution time, the model performs well in comparison to the other models.
However, recall again that the LP time of RAP-MT is highest among the models,
and therefore the overall time of RAP-MT is sometimes longer than those of RAP-
MG+ and RAP-F2.
10.7 Concluding Remarks
In this chapter, we have presented a number of compact integer programming models
for two types of closely-related power-optimal trees in ad hoc wireless networks.
We have provided theoretical analysis of the strengths of the models in LP bound.
Experiments on networks of various sizes have been presented to shed light on the
models’ performance in bounding and obtaining integer optimum.
Several concluding remarks can be made from the analysis and experiments.
First, the models vary significantly in their LP strengths. For small networks, this is
not a crucial aspect at all for approaching integer optimum. When the network size
grows, however, the model choice becomes a crucial aspect, and an LP-stronger
model tends to be more efficient in enabling optimal or close-optimal solutions.
Second, the experimental results illustrate the trade-off between the time required
for bounding and the size of the enumeration tree. For the two types of optimal trees
considered in the chapter, the LP tightness pays off in terms of scalability. Third, for
some models, valid inequalities are very useful for enhancing the performance, and
sometimes a weak model combined with valid inequalities may be a better choice
than a larger model with a tight LP-bound.
There are several directions of complementary investigations. One is a more extensive
study of the multi-tree model, including bounding procedures based on this
model for very large instances. Another topic is variations of RAP-F2 and RAP-
MT, obtained by adapting models for the Steiner tree problem [45], and their performance
study. Structural properties and further developments of solution procedures