Linear Programming & its Applications to Wireless Networks

Guofeng Deng

IMPACT Lab, Arizona State University

G. Deng 2MPACTIArizona State

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

G. Deng 3MPACTIArizona State

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.

G. Deng 4MPACTIArizona State

App1: Maximizing Broadcast Lifetime using Multiple Trees

K

iEp

ts

L

iK

i

K

,0

,

:..

max*̂Objective function:

Constraint 1:

Constraint 2:

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

Tree/Node 1 2 3 4 5 Res

A 10 5 12 0 9 (A)

B 9 0 13 5 0 (B)

C 12 6 0 6 10 (C)

Battery Cap 100 220 150 310 160

G. Deng 5MPACTIArizona State

App2: Bounding the Lifetime of Sensor Networks

B123

3/11 + 5/11

3/11

3/11

5/11

3/11 + 3/11

Bhardwaj & Chandrakasan, Bounding the Lifetime of Sensor Networks Via Optimal Role Assignments, INFOCOM’02

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 N+1 is the destination.- t: lifetime; e_i: battery capacity of node IComment: - 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.

G. Deng 6MPACTIArizona State

App3: Multicommodity Flow

Chang & Tassiulas, Energy conserving routing in wireless ad-hoc networks, INFOCOM’00Chang & Tassiulas, Maximum lifetime routing in wireless sensor networks, TON, Vol.12 No.4, 2004Sanka & Liu, Maximum lifetime routing in wireless ad-hoc networks, INFOCOM’04

G. Deng 7MPACTIArizona State

App4: EE Routing in Disaster Recovery Networks

Zussman & Segall, Energy efficient routing in ad hoc disaster recovery networks, Ad Hoc Networks, Vol.1, 2003

\bar{f}_{i,j}: the amount of info transmitted from i to j until time TR: receiver nodesd: destinationr_i: the ratio between the rate inwhich info is generated at badge node i and the maximum possible flow on a link connecting smart badges

G. Deng 8MPACTIArizona State

App5: Cross-Layer Design for Lifetime Maximization

Madan et al., Cross-layer design for lifetime maximization in interference-limited wireless ad hoc networks, INFOCOM’05Madan et al., Cross-layer design for lifetime maximization in interference-limited wireless ad hoc networks, IEEE trans. Wireless Communications, Vol.5 No.11, 2006

non-convex!

Tv: node lifetimeN: number of slotsr^n_k: trans rate over link k per unit bandwidth in slot nP^n_k: trans power over link k in slot nP^{max}: maximum trans pwrN_0: noise power

G. Deng 9MPACTIArizona State

App6: Minimum Power Broadcast Tree

Das et al., Minimum Power Broadcast Trees for Wireless Networks: Integer Programming Formulations, INFOCOM’03

3

41

6

8

72

5

Power matrix

Reward matrixR_mn(p)=1 if P_mp ≤ P_mn

Variables Y_i: power of node iX_ij: =1 if there is a explicit link from i to jX_ijk: =1 if the kth transmission is i to j

actual trans implicit trans

Defines relation between continuous and binary variables

Source node has to transmit to at least one other node

Non-source node at most transmits to one other node

Defines relation between X_ij and X_ijk.

Source has to transmit in the 1st step.

Non-source node is not allowed to transmit in the 1st step.

A non-source node is not allowed to transmit until it is reached actually or implicitly.

Source has to transmit in the 1st step.

Each node has to be reached ultimately.

At most one transmission in each step.