Constraint Programming II

1
Constraint Programming II

• March 15, 2001

Martin Henz, School of Computing,
NUS www.comp.nus.edu.sg/henz
2
Today

• Search Components in OPL
• Case study ACC 97/98 Basketball
• Constraint programming techniques

3
Review

• Constraint programming is a framework for
  integrating three families of algorithms
• Propagation algorithms
• Branching algorithms
• Exploration algorithms

4
S E N D M O R E M O N E Y
S ? 9 E ? 4..7 N ? 5..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E 4
E ? 4

S ? 9 E ? 5..7 N ? 6..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E 5
E ? 5
S ? 9 E ? 5 N ? 6 D ? 7 M ? 1 O ? 0 R
? 8 Y ? 2
S ? 9 E ? 6..7 N ? 7..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
E ? 6
E 6


5
Using OPL Syntax
enum Letter S,E,N,D,M,O,R,Y var int lLetter
in 0..9 solve alldifferent(l) lS ltgt 0
lM ltgt 0 lS1000 lE100
lN10 lD lM1000
lO100 lR10 lE lM10000
lO1000 lN100 lE10 lY search
forall(i in Letter ordered by increasing
dsize(li)) tryall(v in 0..9)
li v ?All Solutions Execution ? Run
6
Today

• Search Components in OPL
• Propagation Algorithms
• Programming Branching
• Programming Exploration
• Case study ACC 97/98 Basketball
• Constraint programming techniques

7
Constraint Solving

• Given a satisfiable constraint C and a new
  constraint C.
• Constraint solving is deciding whether C ? C is
  satisfiable.
• Example
• C n gt 2
• C an bn cn

8
Constraint Solving

• Clearly, constraint solving is not possible for
  general constraints.
• Constraint programming separates constraints into
• Basic constraints constraint solving
• Non-basic constraints propagation (incomplete)

9
Basic Constraints in Constraint Programming

• Basic constraints are conjunctions of constraints
  of the form X ? S, where S is a finite set of
  integers.
• Constraint solving is done by intersecting
  domains.
• Example
• C X?1..10, Y?9..20,
• C X?9..15, Y?14..30.
• In practice, we keep a solved form, storing the
  current domain of every variable.

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Basic Constraints and Propagators
1000S 100E 10N D
1000M 100O 10R E 10000M 1000O
100N 10E Y
all different(S,E,N,D, M,O,R,Y)
S ? 1..9 E ? 0..9 N ? 0..9 D ? 0..9 M ?
1..9 O ? 0..9 R ? 0..9 Y ? 0..9
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Basic Constraints and Propagators
1000S 100E 10N D
1000M 100O 10R E 10000M 1000O
100N 10E Y
all different(S,E,N,D, M,O,R,Y)
S ? 1..9 E ? 0..9 N ? 0..9 D ? 0..9 M ?
1 O ? 0..9 R ? 0..9 Y ? 0..9
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Basic Constraints and Propagators
1000S 100E 10N D
1000M 100O 10R E 10000M 1000O
100N 10E Y
all different(S,E,N,D, M,O,R,Y)
S ? 2..9 E ? 0,2..9 N ? 0,2..9 D ?
0,2..9 M ? 1 O ? 0,2..9 R ? 0,2..9 Y ?
0,2..9
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Basic Constraints and Propagators
1000S 100E 10N D
1000M 100O 10R E 10000M 1000O
100N 10E Y
all different(S,E,N,D, M,O,R,Y)
S ? 2..9 E ? 0,2..9 N ? 0,2..9 D ?
0,2..9 M ? 1 O ? 0 R ? 0,2..9 Y ? 0,2..9
and so on and so on
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Basic Constraints and Propagators
1000S 100E 10N D
1000M 100O 10R E 10000M 1000O
100N 10E Y
all different(S,E,N,D, M,O,R,Y)
S ? 9 E ? 5..7 N ? 6..8 D ? 2..8 M ?
1 O ? 0 R ? 2..8 Y ? 2..8
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Completeness of Propagation

• Given Basic constraint C and propagator P.
• Propagation is complete, if for every variable x
  and every value v in the domain of x, there is an
  assignment in which xv that satisfies C and P.
• Complete propagation is also called
• domain-consistency or arc-consistency.

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All Different Example 1

• C v ? 1,2
• w ? 1,2,3,4,5
• x ? 1,2,3,4,5
• y ? 1,2,3
• z ? 4,5
• P alldifferent(w,x,y,z)

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All Different Example 2

• C v ? 1
• w ? 1,2,3,4,5
• x ? 1,2,3,4,5
• y ? 1,2,3
• z ? 5
• P alldifferent(w,x,y,z)

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All Different Example 2

• C v ? 1
• w ? 1,2,3,4,5
• x ? 1,2,3,4,5
• y ? 1,2,3
• z ? 5
• P alldifferent(w,x,y,z) onValue

19
All Different Example 3

• C v ? 1,2,3,4,5
• w ? 1,2,3,4,5
• x ? 1,2,3
• y ? 1,2,3
• z ? 1,2,3
• P alldifferent(w,x,y,z) onValue

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All Different Example 3

• C v ? 1,2,3,4,5
• w ? 1,2,3,4,5
• x ? 1,2,3
• y ? 1,2,3
• z ? 1,2,3
• P alldifferent(w,x,y,z) onDomain

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All Different Example 3

• C v ? 1,2,3,4,5
• w ? 1,2,3,4,5
• x ? 1,2,3
• y ? 1,2,3
• z ? 1,2,3
• P alldifferent(w,x,y,z) onDomain

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Complete All Different Constraint
v
1
w
2
x
3
y
4
z
5
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Complete All Different Constraint
Remove all edges that do not belong to a maximum
matching in bipartite graph Regin 94
v
1
w
2
x
3
y
4
z
5
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Complete All Different Constraint
v
1

• ...by applying
• maximum matching algorithm in bipartite graphs
• Hopcroft,
• Karp 1973
• O(X2 dmax2)

w
2
x
3
y
4
z
5
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Today

• Search Components in OPL
• Propagation Algorithms
• Programming Branching
• Programming Exploration
• Case study ACC 97/98 Basketball
• Constraint programming techniques

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Review Branching for MONEY
enum Letter S,E,N,D,M,O,R,Y var int lLetter
in 0..9 solve alldifferent(l) lS ltgt 0
lM ltgt 0 lS1000 lE100
lN10 lD lM1000
lO100 lR10 lE lM10000
lO1000 lN100 lE10 lY search
forall(i in Letter ordered by
increasing dsize(li)) tryall(v in 0..9)
li v ?All Solutions Execution ?
Run
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Domain Splitting in OPL
search forall(i in Letter) while not
bound(li) do let m dmid(li) in
try li lt m li gt m
endtry
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Today

• Search Components in OPL
• Propagation Algorithms
• Programming Branching
• Programming Exploration
• Case study ACC 97/98 Basketball
• Constraint programming techniques

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Optimization in OPL
enum Letter S,E,N,D,M,O,T,Y var int lLetter
in 0..9 maximize money subject to money
lM10000lO1000lN100lE10lY
alldifferent(l) lS ltgt 0 lM ltgt 0
lS1000 lE100 lN10 lD
lM1000 lO100 lS10
lT lM10000 lO1000 lN100
lE10 lY search forall(i in Letter
ordered by increasing dsize(li)) tryall(v
in 0..9) li v ?All Solutions
Execution ? Run
MOST MONEY
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Review Depth-first Search in OPL
enum Letter S,E,N,D,M,O,T,Y var int lLetter
in 0..9 solve alldifferent(l) lS ltgt 0
lM ltgt 0 lS1000 lE100
lN10 lD lM1000
lO100 lS10 lT lM10000
lO1000 lN100 lE10 lY search
forall(i in Letter ordered by increasing
dsize(li)) tryall(v in 0..9)
li v If nothing else specified,
depth-first search is used.
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Built-in Explorations in OPL

• Depth-first search (OPL default)
• Best-first search
• choose the node first that has maximal (minimal)
  lower bound for optimization function
• OPL BFSearch(money)
• Limited discrepancy search
• explore search tree in order of increasing
  discrepancies
• OPL LDSearch(4)

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Best-first Search for Money
enum Letter S,E,N,D,M,O,T,Y var int lLetter
in 0..9 maximize money subject to money
lM10000lO1000lN100lE10lY
alldifferent(l) lS ltgt 0 lM ltgt 0
lS1000 lE100 lN10 lD
lM1000 lO100 lS10
lT lM10000 lO1000 lN100
lE10 lY search BFSearch(money)
forall(i in Letter ordered by increasing
dsize(li)) tryall(v in 0..9)
li v
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Programming Explorations in OPL

• Works on nodes of search tree
• Uses priority queue for order of nodes
• Parameterized by user-defined components

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The Exploration Algorithm of OPL

• Boolean explore(PriorityQueue Q)
• if Q.empty() return false
• else Node a Q.pop()
• return exploreActiveNode(a,Q)
• Boolean exploreActiveNode a, PriorityQueue Q)
• if a.isFailNode() return
  explore(Q)
• else if a.isLeafNode() return true
• else if a.mustBePostponed(Q) Q.insert(a)
• return
  explore(Q)
• else
  Q.insert(a.getRight())
• a
  a.getLeft()
• return
  exploreActiveNode
• (a,Q)

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Example Implementing Depth-First

• SearchStrategy myDFS()
• evaluated to - OplSystem.getDepth()
• postponed when
• OplSystem.getEvaluation()
• gt
• OplSystem.getBestEvaluation()

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Applying User-defined Exploration
enum Letter S,E,N,D,M,O,R,Y var int lLetter
in 0..9 solve alldifferent(l) lS ltgt 0
lM ltgt 0 lS1000 lE100
lN10 lD lM1000
lO100 lR10 lE lM10000
lO1000 lN100 lE10 lY search
applyStrategy myDFS() forall(i in Letter
ordered by increasing dsize(li))
tryall(v in 0..9) li v
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Combining Branching and Exploration
search applyStrategy myDFS() forall(i in
1..3) ordered by increasing dsize(li))
tryall(v in 0..9) li v
LDSearch(4) forall(i in 4..8) tryall(v
in 0..9) li v
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Today

• Search Components in OPL
• Case study ACC 97/98 Basketball
• Constraint programming techniques

39
ACC 1997/98 A Success Story of Constraint
Programming

• Integer programming enumeration, 24 hours
• Nemhauser, Trick Scheduling a Major College
  Basketball Conference, Operations Research, 1998,
  46(1)
• Constraint programming, less than 1 minute.
• Henz Scheduling a Major College Basketball
  Conference - Revisited, Operations Research,
  2001,
• 49(1)

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Round Robin Tournament Planning Problems

• n teams, each playing a fixed number of times r
  against every other team
• r 1 single, r 2 double round robin.
• Each match is home match for one and away match
  for the other
• Dense round robin
• At each date, each team plays at most once.
• The number of dates is minimal.

41
The ACC 1997/98 Problem

• 9 teams participate in tournament
• Dense double round robin
• there are 2 9 dates
• at each date, each team plays either home, away
  or has a bye
• Alternating weekday and weekend matches

42
The ACC 1997/98 Problem (contd)

• No team can play away on both last dates.
• No team may have more than two away matches in a
  row.
• No team may have more than two home matches in a
  row.
• No team may have more than three away matches or
  byes in a row.
• No team may have more than four home matches or
  byes in a row.

43
The ACC 1997/98 Problem (contd)

• Of the weekends, each team plays four at home,
  four away, and one bye.
• Each team must have home matches or byes at least
  on two of the first five weekends.
• Every team except FSU has a traditional rival.
  The rival pairs are Clem-GT, Duke-UNC, UMD-UVA
  and NCSt-Wake. In the last date, every team
  except FSU plays against its rival, unless it
  plays against FSU or has a bye.

44
The ACC 1997/98 Problem (contd)

• The following pairings must occur at least once
  in dates 11 to 18 Duke-GT, Duke-Wake, GT-UNC,
  UNC-Wake.
• No team plays in two consecutive dates away
  against Duke and UNC. No team plays in three
  consecutive dates against Duke UNC and Wake.
• UNC plays Duke in last date and date 11.
• UNC plays Clem in the second date.
• Duke has bye in the first date 16.

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The ACC 1997/98 Problem (contd)

• Wake does not play home in date 17.
• Wake has a bye in the first date.
• Clem, Duke, UMD and Wake do not play away in the
  last date.
• Clem, FSU, GT and Wake do not play away in the
  fist date.
• Neither FSU nor NCSt have a bye in the last date.
  
• UNC does not have a bye in the first date.

46
Nemhauser/Trick Solution

• Enumerate home/away/bye patterns
• explicit enumeration (very fast)
• Compute pattern sets
• integer programming (below 1 minute)
• Compute abstract schedules
• integer programming (several minutes)
• Compute concrete schedules
• explicit enumeration (approx. 24 hours)
• Schreuder, Combinatorial Aspects of Construction
  of Competition Dutch Football Leagues, Discr.
  Appl. Math, 35301-312, 1992.

47
Modeling ACC 97/98 as Constraint Satisfaction
Problem

• Variables
• 9 18 variables taking values from 0,1 that
  express which team plays home when. Example
  HUNC, 51 means UNC plays home on date 5.
• away, bye similar, e.g. AUNC, 5 or BUNC, 5
• 9 18 variables taking values from 0,1,...,9
  that express against which team which other team
  plays. Example ?UNC, 5 1 means UNC plays team 1
  (Clem) on date 5

48
Modeling ACC 97/97 as Constraint Satisfaction
Problem (contd)

• Constraints
• Example No team plays away on both last dates.
• AClem,17 AClem,18 lt 2, ADuke,17 ADuke,18 lt
  2, ...
• All constraints can be easily formalized in this
  manner.

49
First Step Use Nemhauser/Trick Idea

• Constraint programming for generating all
  patterns.
• CSP representation straightforward.
• computing time below 1 second (Pentium II,
  233MHz)
• Constraint programming for generating all pattern
  sets.
• CSP representation straightforward.
• computing time 3.1 seconds

50
Back to Schreuder

• Constraint programming for abstract schedules
• Introduce variable matrix for abstract
  opponents similar to ? in naïve model
• there are many abstract schedules
• runtime over 20 minutes
• Constraint programming for concrete schedules
• model somewhat complicated, using two levels of
  the element constraint
• runtime several minutes

51
Cains Model

• Alternative to last two phases of Nemhauser/Trick
• Assign teams to patterns in a given pattern set.
• Assign opponent teams for each team and date.
• W.O. Cain, Jr, The computer-assisted heuristic
  approach used to schedule the major league
  baseball clubs, Optimal Strategies in Sports,
  North-Holland, 1977

52
Cain 1977
Schreuder 1992
53
Cains Model in CP main idea

• Remember matrices H, A, B represent patterns and
  matrix ? represents opponents
• Given matrices H, A, B of 0/1 values
  representing given pattern set.
• Vector p of variables ranging from 1 to n pGT
  identifies the pattern for team GT.
• Team GT plays according to pattern indicated by p
  on date 2 HpGT,2HGT,2
• Implemented element(pGT,H2 , HGT,2 )

54
Using Cains Model in CP

• Model for Cain simpler than model for Schreuder
• Runtimes
• patterns to teams 33 seconds
• opponent team assignment 20.7 seconds
• overall runtime for all 179 solutions 57.1
  seconds

55
Friar Tuck

• Constraint programming tool for sport scheduling,
  ACC 97/98 just one instance
• Convenient entry of constraints through GUI
• Friar Tuck is distributed under GPL
• Friar Tuck 1.1 available for Unix and Windows
  95/98 (www.comp.nus.edu.sg/henz/projects/FriarTuc
  k)
• Implementation language Oz using Mozart (see
  www.mozart-oz.org)

56
Today

• Search Strategies in OPL
• Case study ACC 97/98 Basketball
• Constraint programming techniques

57
Constraint Programming Techniques

• Symmetry Breaking
• Redundant Constraints
• Global Constraints

58
Symmetry Breaking

• Often, the most efficient model admits
• many different solutions that are essentially
• the same (symmetric to each other).
• Symmetry breaking tries to improve the
• performance of search by eliminating
• such symmetries.

59
Redundant Constraints

• Pruning of original model is often not sufficient
• Adding redundant constraints sometimes helps

60
Example Grocery Puzzle

• A kid goes into a grocery store and buys four
  items. The cashier charges 7.11, the kid pays
  and is about to leave when the cashier calls the
  kid back, and says Hold on, I multiplied the
  four items instead of adding them Ill try
  again Gosh, with adding them the price still
  comes to 7.11! What were the prices of the four
  items?

61
Model for Grocery Puzzle

• Variables A, B, C, D represent prices of items in
  cents.
• Constraints
• A B C D 711
• A B C D 711 100 100 100

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Additional Constraints

• Redundant constraint
• Symmetries

79 is prime factor of 711. Thus without loss of
generality A divisible by 79
B ? C ? D
63
Solution to Grocery Puzzle
Grocery
Grocery plus Redundancy
Grocery plus Redundancy plus Symmetry
64
Example Fractions

• Find distinct non-zero digits such that the
  following equation holds
• A D G
• 1
• B C E F H I

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Model for Fractions

• One variable for each letter, similar to MONEY
• Constraint
• AEFHI DBCHI GBCEF
• BCEFHI

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Additional Constraints
A D G ? ? B C E F H I

• Symmetries
• Redundant constraints

A 3 ? 1 B C
G 3 ? 1 H I
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Fractions
Fractions plus Symmetries
Fractions plus Symmetries plus Redundancies

• Constraint
• AEFHI
• DBCHI
• GBCEF BCEFHI
• Symmetries
• AEF gt DBC
• DHI gt GEF
• Redundant Constraints
• 3A gt BC
• 3G lt HI

68
Redundant Constraints

• Adding redundant constraints sometimes results
  in dramatic performance improvements.

69
Performance of Symmetry Breaking

• All solution search Symmetry breaking usually
  improves performance often dramatically
• One solution search Symmetry breaking may or may
  not improve performance

70
Global Constraints Hamiltonian Path

• range ChessBoard 1..64
• ChessBoard Knightmove i in ChessBoard
• j j in ChessBoard ...
• var ChessBoard jumpChessBoard
• solve
• forall(p in ChessBoard)
• jumpp in Knightmovep
• circuit(jump)
• forall(p in ChessBoard)
• sum(c in Knightmovep) (jumpc p) 1
  

Hamilton
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Assessment Dont Use It!

• Dont use constraint programming for
• Problems for which there are known efficient
  algorithms or heuristics. Example Traveling
  salesman.
• Problems for which integer programming works
  well. Example Many discrete assignment problems.
• Problems with weak constraints and a complex
  optimization function. Example Timetabling
  problems.

72
Assessment Do Use It!

• Use constraint programming for
• Problems for which integer programming does not
  work (linear models too large).
• Problems for which there are no efficient
  solutions available.
• Problems with tight constraints, where
  propagation can be employed. Example ACC 97/98.
• Problems for which strong branching algorithms
  exist. Example Scheduling with unary resources.

73
Myths Debunked

• A fad! Constraint programming has been used
  successfully in a number of application areas,
  most spectacularly in scheduling
• Universal! More failure stories than success
  stories.
• All new! Many ideas come from AI search and
  Operations Research. In particular, the important
  ideas for scheduling come from OR.
• Artificial Intelligence! All quite earthly
  algorithms. Constraint programming systems
  provide framework for different kinds of
  algorithms to interact.

74
Perspective

• Constraint programming will be added to the OR
  tool set as a standard technique for solving
  combinatorial problems, along with local search.
• Constraint programming techniques will be tightly
  integrated with integer programming and local
  search.