Wireless Network Design: Optimization Models and Solution ...

226 Dag Haugland and Di Yuan
scheme for solving the model, and provided an analysis of two approximation algorithms.
In [6], the authors give, in addition to a more extensive presentation of their
work in [5], a polynomial time approximation scheme. They also commented on the
approximability of SRAP. Das et al. [29] presented an integer programming model
based on network flow. Although the problem addressed in [29] assumes that nodes
have sectored antennas instead of omni-directional antennas, the application of the
model to RAP is straightforward. Using sub-tour elimination for ensuring connectivity,
the authors of [13] computationally compared, for relatively small networks,
three alternatives of variable definition to represent the total power consumption: a
set of binary variables, one continuous power variable per node, and a single continuous
variable for the entire network. The first two alternatives will be examined
in more detail in Sections 10.3.1 and 10.4. In [39], Montemanni and Gambardella
proposed two integer programming models based on network flow and graph cuts,
respectively, and compare solution approaches using these models to that in [5]. An
overview of the integer programming models and solution procedures in [5, 29, 39]
and comparative numerical results are provided in [40]. Note that, in [40] and some
other references, ARAP is referred to as the minimum power symmetric connectivity
problem. In Sections 10.4 and 10.5, we review some flow-based models in
[29, 40], and present new compact models.
10.2.4 Optimal Trees and Arborescence: Illustrative Example
In this section, we give an example illustrating optimal trees and arborescences as
defined by problems MET and RAP, respectively, and compare them with optimal
solutions to related well-studied problems. Since the related problems are defined in
terms of cost minimization, we let cost and power be synonymous in this context.
If the cost parameters were associated with links instead of nodes, MET and
RAP become classical optimal arborescences and trees. Specifically, MEBT, MEMT,
ARAP, and SRAP are peers of minimum cost arborescence, Steiner arborescence,
minimum spanning tree, and Steiner tree, respectively. It is instructive to use an example
to illustrate the optimal solutions to MET and RAP, and compare them to the
classical optimal tree and arborescence solutions. Figure 1(a) shows the topology of
a network of 10 nodes. The power parameters are displayed in Figure 1(b). Note
that, under the assumption of symmetric link costs, minimum cost arborescence
and MST are equivalent (except the specification of link direction in the former),
and have the same optimal value. The observation holds also for optimal Steiner
arborescence and Steiner tree.
Figure 10.2 shows the optimal arborescence of MEBT and that of a group of 5
nodes for MEMT, as well as the corresponding optimal solutions of the classical
minimum arborescence problem and the Steiner arborescence problem. There are
clear differences between the minimum cost arborescence and the optimal MEBT
arborescence. Once arc (4,8) is used in the MEBT solution, the additional powers
needed at node 4 to reach nodes 2,6 and 7 are less than those required by the min-