Constraints and Search

Toby WalshCork Constraint Computation Centre (4C)

tw@4c.ucc.ie

Logic & AR Summer School, 2002

Constraint satisfaction

• Constraint satisfaction problem (CSP) is a triple <V,D,C> where:– V is set of variables– Each X in V has set of values, D_X

• Usually assume finite domain• {true,false}, {red,blue,green}, [0,10], …

– C is set of constraints

Goal: find assignment of values to variables to satisfy all the constraints

Constraint solver

• Tree search– Assign value to variable– Deduce values that must be removed from future/unassigned

variables• Constraint propagation

– If any future variable has no values, backtrack else repeat

• Number of choices– Variable to assign next, value to assign

Some important refinements like nogood learning, non-chronological backtracking, …

Constraint propagation

• Enfrocing arc-consistency (AC)– A binary constraint r(X1,X2) is AC iff

for every value for X1, there is a consistent value (often called support) for X2 and vice versa

E.g. With 0/1 domains and the constraint X1 =/= X2Value 0 for X1 is supported by value 1 for X2Value 1 for X1 is supported by value 0 for X2…

– A problem is AC iff every constraint is AC

Tree search

• Backtracking (BT)

• Forward checking (FC)

• Maintaining arc-consistency (MAC)

• Limited discrepany search (LDS)

• Non-chronological backtracking & learning

Modelling case study: orthogonal Latin squares

Or constraint programming isn’t purely declarative!

Modelling decisions

• Many different ways to model even simple problems– It’s not pure declarative programming!

• Combining models can be effective– Channel between models

• Need additional constraints– Symmetry breaking– Implied (but logically) redundant

Orthogonal Latin squares

• Find a pair of Latin squares– Every cell has a different

pair of elements

• Generalized form:– Find a set of m Latin

squares

– Each possible pair is orthogonal

Orthogonal Latin squares

1 2 3 4 1 2 3 42 1 4 3 3 4 1 23 4 1 2 4 3 2 14 3 2 1 2 1 4 3 11 22 33 44 23 14 41 32 34 43 12 21 42 31 24 13

• Two 4 by 4 Latin squares

• No pair is repeated

History of (orthogonal) Latin squares

• Introduced by Euler in 1783– Also called Graeco-Latin or Euler squares

• No orthogonal Latin square of order 2– There are only 2 (non)-isomorphic Latin

squares of order 2 and they are not orthogonal

History of (orthogonal) Latin squares

• Euler conjectured in 1783 that there are no orthogonal Latin squares of order 4n+2– Constructions exist for 4n and for 2n+1– Took till 1900 to show conjecture for n=1– Took till 1960 to show false for all n>1

• 6 by 6 problem also known as the 36 officer problem“… Can a delegation of six regiments, each of which sends a colonel,

a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular 6 by 6 array such that no row or column duplicates a rank or a regiment?”

More background

• Lam’s problem– Existence of finite projective plane of order 10– Equivalent to set of 9 mutually orthogonal Latin

squares of order 10– In 1989, this was shown not to be possible after 2000

hours on a Cray (and some major maths)

• Orthogonal Latin squares are used in experimental design– To ensure no dependency between independent

variables

A simple 0/1 model

• Suitable for integer programming– Xijkl = 1 if pair (i,j) is in row k column l, 0 otherwise– Avoiding advice never to use more than 3 subscripts!

• Constraints– Each row contains one number in each square

Sum_jl Xijkl = 1 Sum_il Xijkl = 1

– Each col contains one number in each squareSum_jk Xijkl = 1 Sum_ik Xijkl = 1

A simple 0/1 model

• Additional constraints– Every pair of numbers occurs exactly once

Sum_kl Xijkl = 1

– Every cell contains exactly one pair of numbersSum_ij Xijkl = 1

Is there any symmetry?

Symmetry removal

• Important for solving CSPs– Especially for proofs of optimality?

• Orthogonal Latin square has lots of symmetry– Permute the rows– Permute the cols– Permute the numbers 1 to n in each square

• How can we eliminate such symmetry?

Symmetry removal

• Fix first row 11 22 33 …

• Fix first column112332..

• Eliminates all this symmetry?

What about a CSP model?

• Exploit large finite domains possible in CSPs– Reduce number of variables

– O(n^4) -> ?

• Exploit non-binary constraints– Problem states that squares contain pairs that are all

different

– All-different is a non-binary constraint our solvers can reason with efficiently

CSP model

• 2 sets of variables– Skl = i if the 1st element in row k col l is i– Tkl = j if the 2nd element in row k col l is j

• How do we specify all pairs are different?– All distinct (k,l), (k’,l’) if Skl = i and Tkl = j then Sk’l’=/ i or Tk’l’ =/ j

O(n^4) loose constraints, little constraint propagation!

What can we do?

CSP model

• Introduce auxiliary variables– Fewer constraints, O(n^2)– Tightens constraint graph => more propagation– Pkl = i*n + j if row k col l contains the pair i,j

• Constraints– 2n all-different constraints on Skl, and on Tkl– All-different constraint on Pkl– Channelling constraint to link Pkl to Skl and Tkl

CSP model v O/1 model

• CSP model– 3n^2 variables

– Domains of size n, n and n^2+n

– O(n^2) constraints

– Large and tight non-binary constraints

• 0/1 model– n^4 variables

– Domains of size 2

– O(n^4) constraints

– Loose but linear constraints

• Use IP solver!

Solving choices for CSP model

• Variables to assign– Skl and Tkl, or Pkl?

• Variable and value ordering

• How to treat all-different constraint– GAC using Regin’s algorithm O(n^4)– AC using the binary decomposition

Good choices for the CSP model

• Experience and small instances suggest:– Assign the Skl and Tkl variables– Choose variable to assign with Fail First

(smallest domain) heuristic• Break ties by alternating between Skl and Tkl

– Use GAC on all-different constraints for Skl and Tkl

– Use AC on binary decomposition of large all-different constraint on Pkl

Performancen 0-1 model

Fails t/sec

CSP model AC

Fails t/sec

CSP model GAC

Fails t/sec

4 4 0.11 2 0.18 2 0.38

5 1950 4.05 295 1.39 190 1.55

6 ? ? 640235 657 442059 773

7* 20083 59.8 91687 51.1 57495 66.1

General methodology?

• Choose a basic model

• Consider auxiliary variables– To reduce number of

constraints, improve propagation

• Consider combined models– Channel between views

• Break symmetries

• Add implied constraints– To improve propagation

2ns case study: Langford’s problem

Langford’s problem

• Prob024 @ www.csplib.org

• Find a sequence of 8 numbers– Each number [1,4]

occurs twice– Two occurrences of i

are i numbers apart

• Unique solution– 41312432

Langford’s problem

• L(k,n) problem– To find a sequence of k*n

numbers [1,n]

– Each of the k successive occrrences of i are i apart

– We just saw L(2,4)

• Due to the mathematician Dudley Langford– Watched his son build a

tower which solved L(2,3)

Langford’s problem

• L(2,3) and L(2,4) have unique solutions• L(2,4n) and L(2,4n-1) have solutions

– L(2,4n-2) and L(2,4n-3) do not– Computing all solutions of L(2,19) took 2.5 years!

• L(3,n)– No solutions: 0<n<8, 10<n<17, 20, ..– Solutions: 9,10,17,18,19, ..

A014552Sequence: 0,0,1,1,0,0,26,150,0,0,17792,108144,0,0,39809640,326721800,

0,0,256814891280,2636337861200

Basic model

• What are the variables?

Basic model

• What are the variables?Variable for each occurrence of a number

X11 is 1st occurrence of 1X21 is 1st occurrence of 2..X12 is 2nd occurrence of 1X22 is 2nd occurrence of 2..

• Value is position in the sequence

Basic model

• What are the constraints?– Xij in [1,n*k]– Xij+1 = i+Xij– Alldifferent([X11,..Xn1,X12,..Xn2,..,X1k,..Xnk

])

Recipe• Create a basic model

– Decide on the variables

• Introduce auxiliary variables– For messy/loose constraints

• Consider dual, combined or 0/1 models

• Break symmetry

• Add implied constraints

• Customize solver– Variable, value ordering

Break symmetry

• Does the problem have any symmetry?

Break symmetry

• Does the problem have any symmetry?– Of course, we can invert any sequence!

Break symmetry

• How do we break this symmetry?

Break symmetry

• How do we break this symmetry?– Many possible ways– For example, for L(3,9)

• Either X92 < 14 (2nd occurrence of 9 is in 1st half)

• Or X92=14 and X82<14 (2nd occurrence of 8 is in 1st half)

Recipe• Create a basic model

– Decide on the variables

• Introduce auxiliary variables– For messy/loose constraints

• Consider dual, combined or 0/1 models

• Break symmetry

• Add implied constraints

• Customize solver– Variable, value ordering

What about dual model?

• Can we take a dual view?

What about dual model?

• Can we take a dual view?

• Of course we can, it’s a permutation!

Dual model

• What are the (dual) variables?– Variable for each position i

• What are the values?

Dual model

• What are the (dual) variables?– Variable for each position i

• What are the values?– If use the number at that position, we cannot

use an all-different constraint– Each number occurs not once but k times

Dual model

• What are the (dual) variables?– Variable for each position i

• What are the values?– Solution 1: use values from [1,n*k] with the value

i*n+j standing for the ith occurrence of j– Now want to find a permutation of these numbers

subject to the distance constraint

Dual model

• What are the (dual) variables?– Variable for each position i

• What are the values?– Solution 2: use as values the numbers [1,n]

– Each number occurs exactly k times

– Fortunately, there is a generalization of all-different called the global cardinality constraint (gcc) for this

Global cardinality constraint

• Gcc([X1,..Xn],l,u) enforces values used by Xi to occur between l and u times– All-different([X1,..Xn]) = Gcc([X1,..Xn],1,1)

• Regin’s algorithm enforces GAC on Gcc in O(n^2.d)– Regin’s papers are tough to follow but this

seems to beat his algorithm for all-different!?

Dual model

• What are the constraints?– Gcc([D1,…Dk*n],k,k)– Distance constraints?

Dual model

• What are the constraints?– Gcc([D1,…Dk*n],k,k)– Distance constraints:

• Di=j then Di+j+1=j

Combined model

• Primal and dual variables

• Channelling to link them– What do the channelling constraints look like?

Combined model

• Primal and dual variables

• Channelling to link them– Xij=k implies Dk=i

Solving choices?

• Which variables to assign?– Xij or Di

Solving choices?

• Which variables to assign?– Xij or Di, doesn’t seem to matter

• Which variable ordering heuristic?– Fail First or Lex?

Solving choices?

• Which variables to assign?– Xij or Di, doesn’t seem to matter

• Which variable ordering heuristic?– Fail First very marginally better than Lex

• How to deal with the permutation constraint?– GAC on the all-different– AC on the channelling– AC on the decomposition

Solving choices?

• Which variables to assign?– Xij or Di, doesn’t seem to matter

• Which variable ordering heuristic?– Fail First very marginally better than Lex

• How to deal with the permutation constraint?– AC on the channelling is often best for time

Conclusions

• Modelling is an art but there are patterns– Develop basic model

• Decide on the variables and their values

– Use auxiliary variables to represent constraints compactly/efficiently

– Consider dual, combined and 0/1 models– Break symmetry– Add implied constraints– Customize solver for your model