Linear Programming

1
Linear Programming its Applications to Wireless
Networks

• Guofeng Deng
• IMPACT Lab, Arizona State University

2
Outline

• Linear programming (LP)
• Formulation
• Solutions
• Flow model
• Applications
• Maximizing broadcast lifetime
• Optimal role assignments
• Multicommodity flow
• Energy efficient routing in disaster recovery
  networks
• Cross-layer design for lifetime maximization
• Minimum power broadcast tree

3
LP Summary

• LP
• Linear objective function
• Continuous variables
• Linear constraints (equations or inequalities)
• Solutions
• Simplex methods
• Interior-point methods
• Software tools
• Cplex, GLPK, Matlab
• Beyond LP
• Integer linear programming (ILP) variables are
  integers.
• It is called mixed integer programming (MIP) if
  not all variables are integers.
• The problem becomes NP-hard.
• Approximation methods include branch-and-bound,
  branch-and-cut.
• If removing integer constraints, LP provides a
  lower/upper bound to a minimization/maximization
  problem.
• Nonlinear programming some constraints or the
  objective function is nonlinear.

4
App1 Maximizing Broadcast Lifetime using
Multiple Trees

• Summary
• Problem Given a set of broadcast trees in the
  form of power consumption of each node,
  maximizing broadcast lifetime using multiple
  trees sequentially.
• Variables Duration of each tree being used. We
  assume duration is indefinitely divisible.
• Constraints For each node, the overall amount of
  energy that can be consumed in all the trees is
  limited by its battery capacity.
• Notations
• K a set of broadcast trees
• ?(?) the duration of tree ? ? K
• pi(?) power of node i on tree ?
• Ei battery capacity of node i

Objective function Constraint 1 Constraint
2
5
App2 Bounding the Lifetime of Sensor Networks

• Summary
• Problem Given a pair of source and destination
  nodes and a set of intermediate nodes, maximize
  the lifetime, i.e., the amount of packets that is
  transmitted from source to designation.
• Variables f_ij the flow from i to j.
• Constraints see below.
• Notations
• Node 1 is the source and N1 is the destination.
• t lifetime e_i battery capacity of node I
• Comment
• - The formulation was later extended to
  accommodate multiple source and single sink.

For any intermediate node, which does not
generate any flow, the amount of incoming flow
matches the amount of outgoing flow.
This is the total amount of flow injected to the
network, i.e., the difference between the amount
of flow outgoing from source and that incoming to
source.
Bhardwaj Chandrakasan, Bounding the Lifetime of
Sensor Networks Via Optimal Role Assignments,
INFOCOM02
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App3 Multicommodity Flow
Chang Tassiulas, Energy conserving routing in
wireless ad-hoc networks, INFOCOM00 Chang
Tassiulas, Maximum lifetime routing in wireless
sensor networks, TON, Vol.12 No.4, 2004 Sanka
Liu, Maximum lifetime routing in wireless ad-hoc
networks, INFOCOM04
7
App4 EE Routing in Disaster Recovery Networks
\barf_i,j the amount of info transmitted
from i to j until time T R receiver nodes d
destination r_i the ratio between the rate
in which info is generated at badge node i and
the maximum possible flow on a link connecting
smart badges
Zussman Segall, Energy efficient routing in ad
hoc disaster recovery networks, Ad Hoc Networks,
Vol.1, 2003
8
App5 Cross-Layer Design for Lifetime Maximization
Tv node lifetime N number of slots rn_k trans
rate over link k per unit bandwidth in
slot n Pn_k trans power over link k
in slot n Pmax maximum trans pwr N_0 noise
power
non-convex!
Madan et al., Cross-layer design for lifetime
maximization in interference-limited wireless ad
hoc networks, INFOCOM05 Madan et al.,
Cross-layer design for lifetime maximization in
interference-limited wireless ad hoc networks,
IEEE trans. Wireless Communications, Vol.5 No.11,
2006
9
App6 Minimum Power Broadcast Tree
Defines relation between continuous and binary
variables
Source node has to transmit to at least one other
node
4
6
1
8
Non-source node at most transmits to one other
node
3
5
Source has to transmit in the 1st step.
actual trans
implicit trans
Defines relation between X_ij and X_ijk.
7
2
Variables
A non-source node is not allowed to transmit
until it is reached actually or implicitly.
Non-source node is not allowed to transmit in the
1st step.
Y_i power of node i X_ij 1 if there is a
explicit link from i to j X_ijk 1 if the kth
transmission is i to j
Power matrix
At most one transmission in each step.
Reward matrix R_mn(p)1 if P_mp P_mn
Each node has to be reached ultimately.
Das et al., Minimum Power Broadcast Trees for
Wireless Networks Integer Programming
Formulations, INFOCOM03
Source has to transmit in the 1st step.