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Revisiting the Optimal Scheduling Problem. Sastry Kompella 1 , Jeffrey E. Wieselthier 2 , Anthony Ephremides 3 1 Information Technology Division, Naval Research Laboratory, Washington DC 2 Wieselthier Research, Silver Spring, MD

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revisiting the optimal scheduling problem

Revisiting the Optimal Scheduling Problem

Sastry Kompella1, Jeffrey E. Wieselthier2, Anthony Ephremides3

1 Information Technology Division, Naval Research Laboratory, Washington DC

2 Wieselthier Research, Silver Spring, MD

3 ECE Dept. and Institute for Systems Research, University of Maryland, College Park, MD

CISS 2008 – Princeton University, NJ

March 2008


This work was supported by the Office of Naval Research.

elementary scheduling

= transmission rate

(or “capacity”)

Elementary Scheduling

Demand: bits (volume)





Minimize Schedule Length for given demand

bits/sec (rate)

CISS 2008 2 Princeton University, NJ

elementary scheduling cont

Rate: bits/sec

Elementary Scheduling (cont…)

Volume: bits per frame

Maximize total delivery (rate or volume)

for given schedule length (sec)

LP problems !!

CISS 2008 3 Princeton University, NJ

more generally

= # of subsets of the

set of links ( )

= set of links activated

in slot (duration )


Feasibility of

= rate on link i when

set is activated.

More generally

Also an LP !!

Past work:

Truong, Ephremides

Hajek, Sasaki

Borbash, Ephremides


CISS 2008 4 Princeton University, NJ

more complicated

= channel gain from



= Transmit Power at

More Complicated
  • Incorporation of the physical layer (through SINR)
  • Still an LP problem for given ‘s and ‘s
    • Feasibility criterion on the ‘s
  • But, may also choose either or or both.

CISS 2008 5 Princeton University, NJ

our approach column generation
Our Approach: Column Generation
  • Idea: Selective enumeration
    • Include only link sets that are part of the optimal solution
    • Add new link sets at each iteration
      • Only if it results in performance improvement
  • Implementation details
    • Decompose the problem: Master problem and sub-problem
    • Master problem is LP
    • Sub-problem is MILP
  • Optimality
    • Depends on termination criterion
    • Finite number of link sets
  • Complexity: worst case is exponential
    • Typically much faster

CISS 2008 6 Princeton University, NJ

column generation
Column Generation
  • Master Problem: start with a subset of feasible link sets
  • Sub-problem: generate new feasible link sets
  • Steps
    • Initialize Master problem with a feasible solution
    • Master problem generates cost coefficients (dual multipliers)
    • Sub-problem uses cost coefficients to generate new link sets
    • Master problem receives new link sets and updates cost coefficients
    • Algorithm terminates if can’t find a link set that enables shorter schedule


dual multipliers

new link set


(Column Generator)

CISS 2008 7 Princeton University, NJ

master problem
Master Problem
  • Restricted form of the original problem
    • Subset of link sets used; Initialized with a feasible schedule
      • e.g. TDMA schedule
    • Schedule updated during every iteration
    • Solution provides upper bound (UB) to optimal schedule length
    • Yields cost coefficients for use in sub-problem
      • Solution to dual of master problem

CISS 2008 8 Princeton University, NJ

sub problem 1
Sub-problem (1)
  • How to generate new columns?
    • Idea based on revised simplex algorithm
    • Sub-problem receives dual variables from master problem
    • Sub-problem can compute “reduced costs” based on use of any link set
  • Sub-problem
    • Find the matching that provides the most improvement

CISS 2008 9 Princeton University, NJ

sub problem 2
Sub-problem (2)
  • Mixed-integer linear programming (MILP) problem
  • Algorithm Termination
    • If solution to “MAX” problem provides improved performance
      • Add this column to master problem
        • Will improve the objective function
      • Otherwise, current UB is optimal
    • If lower bound and upper bound are within a pre-specified value

CISS 2008 10 Princeton University, NJ

extend to variable transmit power scenario
Extend to “variable transmit power” scenario
  • Nodes allowed to vary transmit power
  • Sub-problem generates better matchings by reducing cumulative interference
    • More links can be active simultaneously
  • Still a mixed-integer linear programming problem
    • No additional complexity

Sub-problem Constraints

Transmission Constraints

SINR Constraints

CISS 2008 11 Princeton University, NJ

an example
An Example
  • 6-node network, 8 links
  • Fixed transmit power: 22% reduction in schedule length compared to TDMA
  • Variable transmit power: 32% reduction in schedule length compared to TDMA

Fixed transmit Power: schedule length = 124.9 s







TDMA schedule = 159.2 s

Variable transmit power: schedule length = 108.6 s

CISS 2008 12 Princeton University, NJ

15 node network
15-node network

Schedule length for different instances (sec)

Spatial reuse ( = Avg. number of links per matching)

CISS 2008 13 Princeton University, NJ

introducing routing

= # of sessions

= set of links that

originate with node

= source node for


= set of links that

end with node

= destination node

for session

Introducing Routing

Flow Equations:

For each session and for each node

Written concisely,

CISS 2008 14 Princeton University, NJ

  • Multi-path routing between and for each session
  • Still an LP problem
  • Column generation still applies

CISS 2008 15 Princeton University, NJ

15 node network1
15-node network

Variable transmit Power

Fixed transmit Power

CISS 2008 16 Princeton University, NJ

summary conclusions
Summary & Conclusions
  • Physical Layer-aware scheduling
    • LP problem but complex
  • Solution approach based on column generation works
    • Decompose the problem into two easier-to-solve problems
    • Worst-case exponential complexity but much faster in practice
      • Enumeration of feasible link sets a priori is average-case exponential
  • Incorporation of Routing
  • Possibility of Power and Rate control

Makes the MAC issue irrelevant !!

CISS 2008 17 Princeton University, NJ