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
M
2
C2
   i
i
i 1
Ci
1
C1
M
1
CM
3
2
M

Minimize Schedule Length for given demand
Demand:
Ci
= transmission rate
(or “capacity”)
CISS 2008
Vi
bits (volume)
i 
Ri
Vi
Ci
bits/sec (rate)
Ri 
Ci i

 i 
M
M
i 1
i 1
    i   
M
Vi
  
i 1 Ci

2
Ri
Ci
Ri
Ci
M
Ri
 1 (feasibili ty)

i 1 Ci
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Elementary Scheduling (cont…)
Maximize total delivery (rate or volume)
for given schedule length  (sec)
Volume:
Vi
bits per frame
Rate: Ri bits/sec
M
Max
V
i 1
M
i 1
i 1
 Ri  
i
M
s. to.
M
 Max
Vi  

M
C 
i 1
i 1
Ci i
i
M
s. to

i 1
i
i

LP problems !!
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More generally
K
   i
i 1
1
2
3
Schedule S  ( Si , i )

i 1
s. to
Past work:
Truong, Ephremides
Hajek, Sasaki
Borbash, Ephremides
etc
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K = # of subsets of the
set of M links (  2 M)
i  1,..., K Si = set of links activated
in slot i (duration i )
K
Min
K

i
Also an LP !!
C 
j:iS j
i
j
 Vi
i  1,..., M
Feasibility of S j
or
C 
j:iS j
ij
j
 Vi
i  1,..., M
4
Cij = rate on link i when
set S j is activated.
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More Complicated
Pi GTi Di
   Pk GT D
k i
kS j
k
Ti
  ij
e.g.
i
Di
GTi Di = channel gain from
Ti to Di
Pi = Transmit Power at Ti
i
ij   Cij 
link
 (c)  2c /W 1
 Incorporation of the physical layer (through SINR)
 Still an LP problem for given Pi ‘s and Cij ‘s
 Feasibility criterion on the
S ‘s
 But, may also choose either P or C or both.
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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
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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
MASTER PROBLEM
dual multipliers
new link set
SUB-PROBLEM
(Column Generator)
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Master Problem
 Restricted form of the original problem
 Subset of link sets used; Initialized with a feasible schedule
S S
e.g. TDMA schedule
 Schedule S updated during every iteration

 Solution provides upper bound (UB)
to optimal schedule length
 Yields cost coefficients (i )
for use in sub-problem
 Solution to dual of master problem
|S|
Min     j
j 1
s. to
C 
j:iSj
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ij
j
 Vi , i  1,..., M
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Sub-problem (1)
 How to generate new columns?
 Idea based on revised simplex algorithm
 Sub-problem receives dual variables (i ) from master problem
 Sub-problem can compute “reduced costs” based on use of any link set
k
zk  1   i Cik
iS k
 Sub-problem
 Find the matching that provides the most improvement
Max  i Cik
kS \ S
iS k
S \ S - Matchingsin S that are not in S
s. to
feasibilit y constraint s of link sets
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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

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Extend to “variable transmit power” scenario
 Nodes allowed to vary transmit power
Sub-problem Constraints
0  Pi  PMAX
 ( D   GT D PMAX ) xi    GT D Pk 
 Sub-problem generates better
i
matchings by reducing cumulative
interference
i
GTi Di Pi    GTk Di PMAX
 More links can be active
k i
k
i
i  1,..., M
k i
simultaneously
x  x
 Still a mixed-integer linear
iO ( n )
programming problem
 No additional complexity
E (n)
k
k i
i
jE ( n )
j
n  1,..., N
xi  {0,1},
i  1,..., M
0  Pn  PMAX ,
n  1,..., N
Transmission
Constraints
n
1
SINR
Constraints
O (n)
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An Example
Fixed transmit Power: schedule length = 124.9 s
 6-node network, 8 links
1
Matching
1
2
3
4
5
6
7
8
6
3
5
2
4
TDMA schedule = 159.2 s
Active Links
1→2
2→3
3→4
3→6
4→5
5→3
5→6
6→1
Active Links
1 → 2, 5 → 3
2→3
3 → 4, 6 → 1
3→6
3 → 6, 4 → 5
5 → 3, 6 → 1
5→3
5→6
Duration
3.5
58.2
17.5
4.4
13.1
0.3
11.6
16.3
Variable transmit power: schedule length = 108.6 s
Duration
3.5
58.2
17.5
17.5
13.1
15.4
16.3
17.7
Matching
1
2
3
4
5
6
7
Active Links
1 → 2, 3 → 6, 4 → 5
2→3
2 → 3, 6 → 1
3→4
3 → 4, 5 → 6
3→6
5 → 3, 6 → 1
Duration
13.1
55.9
2.3
1.2
16.3
4.4
15.4
 Fixed transmit power: 22% reduction in schedule length compared to TDMA
 Variable transmit power: 32% reduction in schedule length compared to TDMA
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15-node network
Schedule length for different instances (sec)
Links
TDMA
Fixed
Variable transmit
transmit power
power
5
17.5
15.4
15.4
15
34.6
28.9
26.0
25
48.5
33.6
27.8
Spatial reuse ( = Avg. number of links per matching)
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Links
TDMA
Fixed
Variable transmit
transmit power
power
5
1
1.2
1.2
15
1
1.4
1.5
25
1
1.54
1.84
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Introducing Routing
Flow Equations:
For each session l , (1  l  L) and for each node n
n  Tl 
n  Tl

iO ( n )
and n  Dl  
n  Dl 
l
v
 i  Vl
l
v
 i
iO ( n )
 
l
v
 j 0
jE ( n )
l
v
 j  Vl
jE ( n )
n
O (n)
L = # of sessions
O (n) = set of links that
originate with node n
E (n) = set of links that
end with node n
Tl = source node for
session l
Dl = destination node
for session l
Written concisely,
A.v  V
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E (n)
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Formulation
K
Min

i 1
s. to
i
A.v  V
M
 Cij j 
j:iS j
l
v
i
i  1,..., M
1l  L
 Multi-path routing between
Tl and Dl for each session l
 Still an LP problem
 Column generation still applies
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15-node network
Variable transmit Power
Fixed transmit Power
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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 !!
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