Wireless Network Design: Optimization Models and Solution ...

10 Integer Programming Models for Power-optimal Trees in Wireless Networks 225
in [47]. For the MST heuristic, the bound in [8] is in fact tight. For MIP, Wan et
al. [48] showed that the algorithm does not have a constant approximation ratio,
but algorithms having this property can be obtained by adapting several algorithms
developed for the Steiner tree problem.
Since the introduction of MET and the first algorithms BIP and MIP, many approximation
algorithms and heuristics have been proposed in the literature. We refer
to [12, 14, 15, 18, 30, 33, 36, 37, 42, 48, 53, 55], and the references therein,
for additional details. Some of the algorithms [42, 48, 55] solve MET; others are
specifically developed for the broadcast case MEBT. The algorithms differ in how
they construct an arborescence and/or how to manipulate a given arborescence to
seek for improvement.
Integer programming for MET has been investigated in [7, 10, 28, 38, 54]. The
models in the references use network flow, sub-tour elimination, graph cuts, or constraints
of set-covering type with an implicit tree representation, to enforce connectivity
from the source to the destinations. Some comparative studies are reported in
[10, 38]. The integer programming approaches in [7, 54] are specifically designed
for MEBT; the other references deal with the general problem MET. In Section
10.3, we review and compare some compact MET models presented in [10, 28].
10.2.3 Range Assignment
In range assignment, the power vector P must enable a subgraph containing sufficiently
many bi-directional links to strongly connect all nodes in D. Below we give
a formal definition.
[RAP ] Find a power vector (P1,P2,...,PN) ∈ ℜ N + of minimum sum, such that the
induced graph (V,E P ), where E P = {(i, j) ∈ E : pi j ≤ Pi and p ji ≤ Pj},
connects the nodes in D.
Problems ARAP and SRAP correspond to D = V and D ⊂ V , respectively. There
always exists an optimal solution, in which (V,E P ) contains a tree. For ARAP, this
is a spanning tree of graph G. For SRAP, the tree spans all nodes in D and possibly
some Steiner nodes as well.
RAP arises in the context of optimal topology control [46] of wireless networks
by means of power adjustment. Up to date, the literature has focused on the ARAP
problem. The N P-hardness of ARAP in a 3D Euclidean space (with the aforementioned
power formula) was proved by Kirousis et al. [35]. Later, Clementi et
al. [19] provided the 2D-space extension of the N P-hardness result. Polynomialtime
algorithms for the 1D case of ARAP are provided in [21, 26]. An extension of
ARAP, where the power formula includes a node-specific scaling factor ci,i ∈ V ,
i.e., pi j = ciκd α i j , is studied in [9]. Approximation algorithms and heuristics for
ARAP with a hop limit constraint are provided in [17, 20, 22, 27].
In [5], Althaus et al. presented an integer programming model of ARAP using
sub-tour elimination constraints. The authors presented a constraint generation