1 Lecture 5 Set Packing Problems Set Partitioning Problems

1

Lecture 5 Lecture 5 Set Packing ProblemsSet Packing Problems

Set Partitioning ProblemsSet Partitioning Problems

2

OutlineOutline

set packing problems set packing problems

set partitioning problemsset partitioning problems

3

Set Packing ProblemsSet Packing Problems

4

ContextContext

a set a set S S = {1, 2…, = {1, 2…, mm} }

a collection of subsets of a collection of subsets of SS,, , such that , such that

each subset carry a value each subset carry a value problem: to maximize the total value of problem: to maximize the total value of

subsets selected such that no element is subsets selected such that no element is

selected more than onceselected more than once

5


S S = {1, 2, 3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}

= {{1, 2, 5}, {1, 3}, {2, 4}, {3, 6}, {2, 3, 6}}= {{1, 2, 5}, {1, 3}, {2, 4}, {3, 6}, {2, 3, 6}} examples examples

{1, 2, 5}, {3, 6}: a pack{1, 2, 5}, {3, 6}: a pack

{1, 3}, {2, 4}: a pack{1, 3}, {2, 4}: a pack

{1, 2, 5}, {2, 3, 6}: not a pack{1, 2, 5}, {2, 3, 6}: not a pack

{2, 3, 6}: a pack{2, 3, 6}: a pack

assumption: every subset of value = 1assumption: every subset of value = 1

6


S S = {1, 2, 3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}

= {{1, 2, 5}, {1, 3}, {2, 4}, {3, 6}, {2, 3, 6}}= {{1, 2, 5}, {1, 3}, {2, 4}, {3, 6}, {2, 3, 6}}

: element 1

: element 2

: element 3

: element 4

: element 5

: element 6

1 2 3 4 5

1 2

1 3 5

2 4 5

3

1

2 4 5

max ,

. . 1,

1,

1,

1,

1,

1,

s t

i {0, 1}

set

1: set

2: set

3: set

4: set

5:

Property 9.4: maximization

with all constraints

Property 9.5: all RHS

coefficients = 1

Property 9.6: all matrix

coefficients = 0 or 1

1, if the th member of is in the pack,

0, otherwise.ii

7

Primal-Dual PairPrimal-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])

8

Primal-Dual PairPrimal-Dual Pair

special properties between the primal-dual pairspecial properties between the primal-dual pair

obj function of min obj function of min obj function of max obj function of max

unbounded primal unbounded primal infeasible dual infeasible dual

optimal primal optimal primal optimal dual optimal dual

same objective function valuesame objective function value

easy to deduce the optimal of one from the other easy to deduce the optimal of one from the other

possible to have infeasible primal and infeasible dualpossible to have infeasible primal and infeasible dual

1 1 2 2

11 1 12 2 1 1

1 1 2 2

max ... ,

. .

... ,

... ,

0.

n n

n n

m m mn n m

i

c x c x c x

s t

a x a x a x b

a x a x a x b

x

M M M M M

Primal (Dual)

1 1 2 2

11 1 21 2 1 1

1 1 2 2

min ... ,

. .

... ,

... ,

0.

m m

m n

n n mn n n

j

b y b y b y

s t

a y a y a y c

a y a y a y c

y

M M M M M

Dual (Primal)

9

Comments on Comments on Set Packing ProblemsSet Packing Problems

1 2 3 4 5

1 2

1 3 5

2 4 5

3

1

2 4 5

max ,

. . 1,

1,

1,

1,

1,

1,

s t

i 0

set packing problem with objective coefficients = 1

1 2 3 4 5 6

1 2 5

1 3

2 4

3 6

2 3 6

min ,

. .

1,

1,

1,

1,

1,

s t

i 0

set covering problem with objective coefficients = 1

dual of each other

The two LPs are dual of each other, though the S and in one problem are different from those of the other

problem.

10

Comments on Comments on Set Packing ProblemsSet Packing Problems

similar generalization as in set covering similar generalization as in set covering

problemsproblems weighted set packing problems: RHS weighted set packing problems: RHS

positive integers > 1positive integers > 1

generalized set packing problems: matrix generalized set packing problems: matrix

coefficients = 0 or coefficients = 0 or 1 1

11

Matching Problem: Matching Problem: A Special Type of Set Packing ProblemA Special Type of Set Packing Problem

matching: select the maximum numbers matching: select the maximum numbers

of arcs such that there is no overlapping of arcs such that there is no overlapping

of nodes involved of nodes involved (1, 2), (3, 6), (4, 5)(1, 2), (3, 6), (4, 5)

2

3

15

4

6

12

ExerciseExercise

formulate the matching formulate the matching

problem of the RHS problem of the RHS

network as a set packing network as a set packing

problem problem

2

3

15

4

6

13


14


to cover all the members of to cover all the members of SS by elements of by elements of

without overlapping without overlapping

S S = {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5}

= {{1, 2}, {1, 3, 5}, {2, 4, 5}, {3}, {1}, {4, 5}} = {{1, 2}, {1, 3, 5}, {2, 4, 5}, {3}, {1}, {4, 5}}

e.g., {1, 2}, {3}, and {4, 5} form a partitione.g., {1, 2}, {3}, and {4, 5} form a partition

15


1, if the th member of is in the partition,

0, otherwise.ii

1 2 5

1 3

2 4

3 6

2 3 6

1,

1,

1,

1,

1.

i {0, 1}

either maximization or minimization is all right

S S = {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5}

= {{1, 2}, {1, 3, 5}, {2, 4, 5}, {3}, {1}, {4, 5}} = {{1, 2}, {1, 3, 5}, {2, 4, 5}, {3}, {1}, {4, 5}}

16

Equivalence Between Set Packing Problem Equivalence Between Set Packing Problem and Set Partitioning Problemand Set Partitioning Problem

1 2 3 4 5

1 2

1 3 5

2 4 5

3

1

2 4 5

max ,

. . 1,

1,

1,

1,

1,

1,

s t

i {0, 1}

Set Packing Problem

1 2 3 4 5

1 2 6

1 3 5 7

2 4 5 8

3 9

1 10

2 4 5 11

max ,

. . 1,

1,

1,

1,

1,

1,

s t

i {0, 1}, 1 to 11

Set Partitioning Problem

17


illustrate the transformation of the first illustrate the transformation of the first

equality constraint into an equality constraint into an -constraint-constraint

introduce a dummy variable introduce a dummy variable to the first to the first

constraintconstraint

i {0, 1}


1 2 5

1 3

2 4

3 6

2 3 6

1,

1,

1,

1,

1.

max 1 + 2 + 3 + 4 + 5 + 6,

18


i {0, 1}


1 2 5

1 3

2 4

3 6

2 3 6

1,

1,

1,

1,

1.

max 1 + 2 + 3 + 4 + 5 + 6,

i, {0, 1}

1 2 5

1 3

2 4

3 6

2 3 6

1,

1,

1,

1,

1.

max 1 + 2 + 3 + 4 + 5 + 6 M,

i, {0, 1}

1 2 5

1 3

2 4

3 6

2 3 6

1,

1,

1,

1,

1.

max (M+1)1 + (M+1)2 + 3 + 4 + (M+1) 5 + 6 M,

i, {0, 1}

1 2 5

1 3

2 4

3 6

2 3 6

1,

1,

1,

1,

1.

max (M+1)1 + (M+1)2 + 3 + 4 + (M+1) 5 + 6 M,

19

Further CommentsFurther Comments

set covering problems different from set set covering problems different from set packing problems and set partitioning packing problems and set partitioning problemsproblems

possible to transform a set packing problem possible to transform a set packing problem into a set covering problem, but in general into a set covering problem, but in general not the other way aroundnot the other way around

set covering problem more difficult to solve set covering problem more difficult to solve than the other two problemsthan the other two problems

20

A Simplified A Simplified Air Crew Scheduling ProblemAir Crew Scheduling Problem

21


six flights every day, for cities six flights every day, for cities AA, , BB, and , and

C, C, where where AA is the base is the base8 10 12 181614 2220

A

B

C

leg 1 leg 2leg 3

leg 4

leg 5 leg 6

22


pairings of legs for air crew: pairings of legs for air crew: rules from rules from regulation bodies and unionregulation bodies and union

simplified rules in the examplesimplified rules in the example at most eight hours flying time in a pairingat most eight hours flying time in a pairing

flying time in a pairing = sum of flying times in all legs of the flying time in a pairing = sum of flying times in all legs of the pairingpairing

at most two duties in a pairingat most two duties in a pairing at least nine hours for overnight rest (OR) between at least nine hours for overnight rest (OR) between

dutiesduties

23


cost of a pairing = time away from the cost of a pairing = time away from the

base base flying time of the pairing flying time of the pairing cost of pairing 1 = 36cost of pairing 1 = 3610 = 2610 = 26

24


suppose only considering covering the 6 suppose only considering covering the 6

legs in a day legs in a day

let let xxjj = 1 if the = 1 if the jjth pairing is used, and th pairing is used, and xxjj = =

0 otherwise0 otherwise

25


a set partitioning problema set partitioning problem min 26min 26xx11 + 20 + 20xx22 + 2 + 2xx33 + 26 + 26xx44 + 20 + 20xx55 + 26 + 26xx66, , ss..tt..

xx11 + + xx22 = 1, = 1,

xx11 + + xx66 = 1, = 1,

xx33 + + xx44 + + xx55 + + xx66 = 1, = 1,

xx22 + + xx33 + + xx44 + + xx55 = 1, = 1,

xx22 + + xx55 = 1, = 1,

xx55 + + xx66 = 1, = 1,

xxii {0, 1} {0, 1}

26


The dual of the partitioning problemThe dual of the partitioning problem min min 11 + + 22 + + 33 + + 44 + + 55 + 26 + 2666,,

ss..tt..

11 + + 22 ≤ 26 ≤ 26

11 + + 44 + + 55 ≤ 20 ≤ 20

33 + + 44 ≤ 2≤ 2

33 + + 44 ≤ 26≤ 26

33 + + 44 + + 55 + + 66 ≤ 20≤ 20

22 + + 33 + + 66 ≤ 26≤ 26

27

To Construct a SimplifiedTo Construct a SimplifiedAir Crew Scheduling ProblemAir Crew Scheduling Problem

Bangkok

Singapore

Hong Kong

Taipei

Seoul

3 hr 40min

2 hr 15 min3 hr 45 min

1 hr 30 min

3 hr 10 min2 hr 20 min

28

Six Cities Six Cities

1

4

3

0

2

5

2 hrTokyo

8 am 012 0 flight 9 am 034 5 0 flight

It takes 90 minutes to load supply and passengers It takes 90 minutes to load supply and passengers

before departure (including unloading passengers for before departure (including unloading passengers for

the previous flight leg, if applicable).the previous flight leg, if applicable).

All flights are on exact time.All flights are on exact time.

It takes 30 minutes for a plane to reach the terminal It takes 30 minutes for a plane to reach the terminal

after arrival.after arrival.

It takes 30 minutes to unload passengers and clean up It takes 30 minutes to unload passengers and clean up

a plane at the end of its service. a plane at the end of its service.

29

Assumptions for PlanesAssumptions for Planes

30

Itinerary of Itinerary of the Plane for 0-1-2-0 Tourthe 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 re-loading passengers; loading suppliers

13:20 Leaving Seoul

15:20 Arriving Tokyo

15:50 Parked at Terminal; unloading and re-loading passengers; loading suppliers

17:20 Leaving Tokyo

20:30 Arriving Taipei

21:00 Parked at Terminal; unloading passengers

21:30 Parking overnight

31

ExerciseExercise

give the itinerary of the plane for 0-3-give the itinerary of the plane for 0-3-4-5-0 tour4-5-0 tour

based on the itineraries of the planes based on the itineraries of the planes for the 0-1-2-0 and 0-3-4-5-0 tours, for the 0-1-2-0 and 0-3-4-5-0 tours, construct the requirements for air construct the requirements for air stewards and for pilots stewards and for pilots put down all assumptions madeput down all assumptions made

Documents

1 Lecture 5 Set Packing Problems Set Partitioning Problems