Wireless Network Design: Optimization Models and Solution ...

224 Dag Haugland and Di Yuan
tions of network design problems. For further reading on network flow models, see
the textbook of Ahuja et al. [2].
10.2.2 Minimum Energy Broadcast and Multicast
By the notation in Section 10.2.1, MET can be formally defined as follows:
[MET ] 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) ∈ A : pi j ≤ Pi}, has a path
from s to each t ∈ D \ {s}.
If D = V , we arrive at the broadcast case MEBT, otherwise it is MEMT. In an
optimal solution to MET, (V,E P ) contains an arborescence having s as the root. For
MEMT, the arborescence may use some nodes in V \ D.
Formal analysis of the N P-hardness of MET is provided in [14, 15]. The problem
class was first introduced by Wieselthier et al. [49, 50, 51, 52]. For MEBT,
the authors developed a solution approach, referred to as the broadcast incremental
power (BIP) algorithm, by an adaptation of Prim’s algorithm for MST. For MEMT,
the authors proposed a multicast incremental power (MIP) algorithm that combines
BIP and a pruning phase.
Given that MET is N P-hard, we know that there exists no polynomial algorithm
for MET unless such an algorithm exists for all N P-hard problems, which
in its turn contradicts the well-known ‘P is not N P’ conjecture (see e.g. Fortnow
[31] for a discussion of the P versus N P problem, including failed attempts to
prove the conjecture). Discouraging results on the computational tractability of a
problem often trigger the interest in fast solution approaches with a lower ambition
than finding an optimal solution. If no bound on the distance from optimality can be
found for the output of the method, it is referred to as a heuristic method. Otherwise,
the method is called an approximation algorithm.
More precisely, an approximation algorithm for MET is a polynomial algorithm
that, for all instances of the problem, produces a solution with no more total
power consumption than some constant times the smallest achievable total power
consumption. The constant in question must be common for all instances of the
problem, and the smallest valid constant is called the approximation ratio of the
algorithm. For a more detailed introduction to approximation algorithms, we recommend
Chapter 35 of the textbook of Cormen et al. [25].
It has been proved that for geometric instances, BIP is an approximation algorithm
for MEBT, and theoretical performance analysis has been conducted in several
works. Wan et al. [47] gave [13/3,12] as a range of the approximation ratio of
BIP. In [11], the lower bound is strengthened from 13/3 to approximately 4.6. In
[36], Klasing et al. corrected the upper bound in [47] to 12.15. Navarra [41] improved
the upper bound to 6.33. Ambühl [8] proved that a better upper bound is
6. We note that the bounds in [8, 36] are derived for the MST heuristic (i.e., use
the MST as the solution of MEBT), but these results apply to BIP due to a lemma