Approximation Algorithms for Minimum Energy Transmission Scheduling in Rate and Duty-Cycle Constrained Wireless Networks

We consider a constrained energy optimization called minimum energy scheduling problem (MESP) for a wireless network of
N
users transmitting over
M
time slots, where the constraints arise because of interference between wireless nodes that limits their transmission rates along with load and duty-cycle (on-off) restrictions. Since traditional optimization methods using Lagrange multipliers do not work well and are computationally expensive given the nonconvex constraints, we consider approximation schemes for finding the optimal (minimum energy) transmission schedule by discretizing power levels over the interference channel. First, we show the toughness of approximating MESP for an arbitrary number of users
N
even with a fixed
M
. For any
r
> 0, we demonstrate that there does not exist any (
r
,
r
)-bicriteria approximation for this MESP, unless
P
=
NP
. Conversely, we show that there exist good approximations for MESP with given
N
users transmitting over an arbitrary number of
M
time slots by developing fully polynomial (1,1+¿) approximation schemes (FPAS). For any ¿ > 0, we develop an algorithm for computing the optimal number of discrete power levels per time slot (
O
(1/¿)), and use this to design a (1, 1+¿)-FPAS that consumes no more energy than the optimal while violating each rate constraint by at most a 1+¿-factor. For wireless networks with low-cost transmitters, where nodes are restricted to transmitting at a fixed power over active time slots, we develop a two-factor approximation for finding the optimal fixed transmission power value
P

_opt
that results in the minimum energy schedule.