A Second Look at Constraint Programming

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A Second Look at Constraint Programming

• February 17/18, 2000

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Today

• Fundamentals
• Case study ACC 97/98 Basketball
• Constraint programming techniques

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Thursday/Friday

• Applying constraint programming to job-shop and
  other scheduling problems
• Evaluation of constraint programming

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Review

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

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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


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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
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Today

• Fundamentals
• Case study ACC 97/98 Basketball
• Constraint programming techniques

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Fundamentals

• Constraint solving
• Basic constraints and propagators
• Completeness of propagation

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Constraint Problems

• A constraint problem consists of
• number of variables n
• constraints c1,,cm ? ?n
• The problem is to find
• a (v1,,vn)? ?n such that
• a ? ci , for all 1 ? i ? m

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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

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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)

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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?domc(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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Example Complete All Different

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

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Example Complete All Different

• C w ? 1,2,3,4
• x ? 2,3,4
• y ? 2,3,4
• z ? 2,3,4
• P alldifferent(w,x,y,z)
• Most efficient known algorithm O(X2 dmax2)
• Regin 1994, using graph matching

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Today

• Fundamentals
• Case study ACC 97/98 Basketball
• Constraint programming techniques

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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, to
  appear

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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.

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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

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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.

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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.

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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.

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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.

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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

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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.

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First Step Back to Nemhauser/Trick!

• 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

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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

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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

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Cain 1977
Schreuder 1992
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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 )

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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

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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)

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Today

• Fundamentals
• Case study ACC 97/98 Basketball
• Constraint programming techniques

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Constraint Programming Techniques

• Symmetry Breaking
• Redundant Constraints
• Global Constraints

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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.

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Redundant Constraints

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

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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?

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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
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Solution to Grocery Puzzle
Grocery
Grocery plus Redundancy
Grocery plus Redundancy plus Symmetry
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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

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Redundant Constraints

• Adding redundant constraints sometimes results
  in dramatic performance improvements.

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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

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Global Constraints Euler Tour

• 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
  

Euler
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Today

• Fundamentals
• Case study ACC 97/98 Basketball
• Constraint programming techniques

55
Thursday/Friday

• Applying constraint programming to job-shop and
  other scheduling problems
• Evaluation of constraint programming