Lecture 5
Set Packing Problems
Set Partitioning Problems
1
Outline
 set
packing problems
 set
partitioning problems
2
Set Packing Problems
3
Context
a
set S = {1, 2…, m}
collection of subsets of S, , such that
each subset carry a value
a
 problem:
to maximize the total value of
subsets selected such that no element is
selected more than once
4
Set Packing Problems

S = {1, 2, 3, 4, 5, 6}

 = {{1, 2, 5}, {1, 3}, {2, 4}, {3, 6}, {2, 3, 6}}

examples


{1, 2, 5}, {3, 6}: a pack

{1, 3}, {2, 4}: a pack

{1, 2, 5}, {2, 3, 6}: not a pack

{2, 3, 6}: a pack
assumption: every subset of value = 1
5
Set Packing Problems
1, if the ith member of  is in the pack,
i  
otherwise.
0,
S
= {1, 2, 3, 4, 5, 6}

= {{1, 2, 5}, {1, 3}, {2, 4}, {3, 6}, {2, 3, 6}}
4
3
4
5 ,
5
5
1
2
4
5
 1,
 1,
 1,
 1,
 1,
 1,
: element 1
: element 2
: element 3
: element 4
: element 5
: element 6
set 5:
set 4:
i  {0, 1}
set 3:
Property 9.6:
all matrix
coefficients = 0
or 1
3
3
set 2:
Property 9.5:
all RHS
coefficients = 1
max 1 2
s.t. 1 2
1
2
set 1:
Property 9.4:
maximization
with all 
constraints
6
Primal-Dual Pair
 “An
interesting observation is that the LP
problem associated with a set packing
problem with objective coefficients of 1
is the dual of the LP problem associated
with a set covering problem with
objective coefficients of 1.” (pp 192 of
[7])
7
Primal-Dual Pair
Primal (Dual)
max
s.t.
c1 x1
Dual (Primal)
c2 x2
...
cn xn ,
min
b1 y1
b2 y2
a11 y1
M
a1n y1
 a21 y2
M
 a2 n y2
...
bm ym ,
s.t.
a11 x1  a12 x2
M
M
am1 x1  am 2 x2

...
 a1n xn
M
M
...  amn xn
xi  0.
 b1 ,
M
 bm ,
...
 am1 yn
M
M
...
 amn yn
y j  0.
 c1 ,
M
 cn ,
special properties between the primal-dual pair

obj function of min  obj function of max

unbounded primal  infeasible dual

optimal primal  optimal dual


same objective function value

easy to deduce the optimal of one from the other
possible to have infeasible primal and infeasible dual
8
Comments on
Set Packing Problems
set covering problem with
objective coefficients = 1
set packing problem with
objective coefficients = 1
max 1 2
s.t. 1 2
1
2
3
4
3
4
5 ,
5
5
3
1
2
4
i  0
5
 1,
 1,
 1,
 1,
 1,
 1,
dual of
each other
min  1  2
s.t.
 1  2
1
2
2
 3
 4
 5
 6 ,
 5
 3
 4
3
 3
 6
 6
 1,
 1,
 1,
 1,
 1,
i  0
The two LPs are dual of each other, though the S and  in
one problem are different from those of the other problem.
9
Comments on
Set Packing Problems
 similar
generalization as in set covering
problems
set packing problems: RHS 
positive integers > 1
 weighted
 generalized
set packing problems: matrix
coefficients = 0 or  1
10
Matching Problem:
A Special Type of Set Packing Problem
 matching:
select the maximum numbers
of arcs such that there is no overlapping
of nodes involved
4
2
 (1,
2), (3, 6), (4, 5)
5
1
6
3
11
Exercise
 formulate
the matching
problem of the RHS
network as a set packing
problem
4
2
5
1
6
3
12
Set Partitioning Problems
13
Set Partitioning Problems

to cover all the members of S by elements of 
without overlapping

S = {1, 2, 3, 4, 5}

 = {{1, 2}, {1, 3, 5}, {2, 4, 5}, {3}, {1}, {4, 5}}

e.g., {1, 2}, {3}, and {4, 5} form a partition
14
Set Partitioning Problems

S = {1, 2, 3, 4, 5}

 = {{1, 2}, {1, 3, 5}, {2, 4, 5}, {3}, {1}, {4, 5}}
1, if the ith member of  is in the partition,
i  
otherwise.
0,
either maximization or minimization is all right
1
1
 2
 5
 3
2
2
 4
3
 3
 6
 6
i  {0, 1}
 1,
 1,
 1,
 1,
 1.
15
Equivalence Between Set Packing Problem
and Set Partitioning Problem
Set Partitioning Problem
Set Packing Problem
max 1 2
s.t. 1 2
1
2
3
4
3
4
5
5
3
1
2
4
i  {0, 1}
max 1
s.t. 1
1
5 ,
5
 1,
 1,
 1,
 1,
 1,
 1,
 2
 2
3
3
2

 4
5 ,
4
5
5
4
5
3
1
2
6
7
8
9
10
11
 1,
 1,
 1,
 1,
 1,
 1,
i  {0, 1}, 1 to 11
16
Equivalence Between Set Packing Problem
and Set Partitioning Problem
 illustrate
the transformation of the first
equality constraint into an -constraint
 introduce
constraint
a dummy variable  to the first
Set Partitioning Problem
max 1 + 2 + 3 + 4
 1  2
1
2
2
+ 5 + 6,
 5
 3
 4
3
 3
i  {0, 1}
 6
 6
 1,
 1,
 1,
 1,
 1.
17
Equivalence Between Set Packing Problem
and Set Partitioning Problem
max 1 + 2 + 3 + 4
Set Partitioning Problem
max 1 + 2 + 3 + 4
 1  2
1
2
 5
 3
 4
3
 3
 6
 6
 1,
 1,
 1,
 1,
 1.

 1  2
1
2
2
 5
 3
 4
3
 3
 6
 6
i,  {0, 1}
i  {0, 1}
(M+1)1 + (M+1)2 + 3 + 4
+ (M+1) 5 + 6  M,
 1  2
1
2
2
 5
 3
 4
3
 3
 6
 6
i,  {0, 1}
max
 1,
 1,
 1,
 1,
 1.

max
  1,
 1,
 1,
 1,
 1.

2
+ 5 + 6,
+ 5 + 6  M,
(M+1)1 + (M+1)2 + 3 + 4
+ (M+1) 5 + 6  M,
 1  2
1
2
2
 5
 3
 4
3
 3
 6
 6
i,  {0, 1}
  1,
 1,
 1,
 1,
 1.
18
Further Comments
 set
covering problems different from set
packing problems and set partitioning
problems
 possible
to transform a set packing problem
into a set covering problem, but in general
not the other way around
 set
covering problem more difficult to solve
than the other two problems
19
A Simplified
Air Crew Scheduling Problem
20
A Simplified
Air Crew Scheduling Problem
 six
flights every day, for cities A, B, and C,
where A is the base
8
10
12
14
16
18
20
22
A
B
leg 1
leg 3
leg 5
leg 4
leg 2
leg 6
C
21
A Simplified
Air Crew Scheduling Problem
 pairings
of legs for air crew: rules from
regulation bodies and union

simplified rules in the example

at most eight hours flying time in a pairing

flying time in a pairing = sum of flying times in all legs of the
pairing

at most two duties in a pairing

at least nine hours for overnight rest (OR) between
duties
22
A Simplified
Air Crew Scheduling Problem
 cost
of a pairing = time away from the
base  flying time of the pairing
 cost
of pairing 1 = 3610 = 26
23
A Simplified
Air Crew Scheduling Problem
 suppose
only considering covering the 6
legs in a day
 let
xj = 1 if the jth pairing is used, and xj =
0 otherwise
24
A Simplified
Air Crew Scheduling Problem



a set partitioning problem
min 26x1 + 20x2 + 2x3 + 26x4 + 20x5 + 26x6,
s.t.







x1 + x2
x1
+ x6
x3 + x4 + x5 + x6
x2 + x3 + x4 + x5
x2
+ x5 = 1,
x5 + x6
xi  {0, 1}
= 1,
= 1,
= 1,
= 1,
= 1,
25
A Simplified
Air Crew Scheduling Problem









The dual of the partitioning problem
min 1 + 2 + 3 + 4 + 5 + 266,
s.t.
1 + 2
≤ 26
1 +
4 + 5
≤ 20
3 + 4
≤2
3 + 4
≤ 26
3 + 4 + 5 + 6
≤ 20
2 + 3
+ 6
≤ 26
26
To Construct a Simplified
Air Crew Scheduling Problem
27
Six Cities
Tokyo
Seoul
2 hr
1
2
2 hr 20 min 3 hr 10 min
1 hr 30 min
0
8 am 012 0 flight
9 am 034 5 0 flight
Taipei
3 hr 40min
5 Bangkok
Hong Kong 3
3 hr 45 min
2 hr 15 min
4
Singapore
28
Assumptions for Planes

It takes 90 minutes to load supply and passengers
before departure (including unloading passengers for
the previous flight leg, if applicable).

All flights are on exact time.

It takes 30 minutes for a plane to reach the terminal
after arrival.

It takes 30 minutes to unload passengers and clean up
a plane at the end of its service.
29
Itinerary of
the Plane for 0-1-2-0 Tour
Time
Activities
7:30
Loading
9:00
Leaving Taipei
11:20
Arriving Seoul
11:50
Parked at Terminal; unloading and reloading passengers; loading suppliers
13:20
Leaving Seoul
15:20
Arriving Tokyo
15:50
Parked at Terminal; unloading and reloading passengers; loading suppliers
17:20
Leaving Tokyo
20:30
Arriving Taipei
21:00
Parked at Terminal; unloading
passengers
21:30
Parking overnight
30
Exercise
 give
the itinerary of the plane for 0-34-5-0 tour
 based on the itineraries of the planes
for the 0-1-2-0 and 0-3-4-5-0 tours,
construct the requirements for air
stewards and for pilots
 put
down all assumptions made
31