Modelling & Solving with Constraints Prof. Toby Walsh University College Cork/Uppsala University.

Modelling & SolvingModelling & Solving with Constraints with Constraints

Prof. Toby WalshProf. Toby Walsh

University College Cork/Uppsala UniversityUniversity College Cork/Uppsala University

Overview• Introduction to constraint programming

– Constraint propagation– Backtracking search

• Modelling case studies– Simple recipe

• Solving with constraints– Global constraints– Set variables– Branching heuristics

Resources• Course links

– www.cs.york.ac.uk/~tw/Links/csps/

• Benchmark problems– www.csplib.org

• Constraints solvers– LP based like ECLIPSE, Java based

solvers like JCL, …

Constraint programming• “Dream” of declarative programming

– State the constraints– Solver finds a solution

• Method of choice for many hard combinatorial problems– Scheduling, assignment, routing, …

Constraints are everywhere!

• No meetings before 10am

• Network traffic < 100 Gbytes/sec

• PCB width < 21cm• Salary > 45k Euros…

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

Example CSP• Course timetabling

– Variable for each courseCS101, OS102 ..

– Domain are possible timeswed9am, fri10am, ..

– Constraints:CS101 \= wed9amCapacity constraints: atmost(3,

[OS102,DB103..],wed9am)Lecturer constraints:

alldifferent([CS101,DB103,…])

Constraint optimization• CSP + objective function

– E.g. objective is Profit = Income - Costs

• Find assignment of vals to vars that:– Satisfies constraints– Maximizes (minimizes) objective

• Often solved as sequence of satisfaction problems

Profit > 0, Profit > Ans1, Profit > Ans2, …

Constraint programming v. Constraint logic programming

• Constraints declaratively specify problem– Logic programming natural approach

Assert constraints, call “labelling” strategy (backtracking search predicate)

• Imperative & functional toolkits• C++, Java, CAML, …

Constraints• Constraints are tuples <S,R> where

– S is the scope, [X1,X2, … Xm]• list of variables to which constraint applies

– R is relation specifying allowed values (goods)

• Subset of D_X1 x D_X2 x … x D_Xm• May be specified intensionally or extensionally

Constraints• Extensional specification

– List of goods (or for tight constraints, nogoods)

• Intensional specification– X1 =/= X2– 5*X1 + 6*X2 < X3– alldifferent([X1,X2,X3,X4]), …

Binary v non-binary• Binary constraint

– Scope covers 2 variables– E.g. not-equals constraint: X1 =/= X2.– E.g. ordering constraint: X1 < X2

• Non-binary constraint– Scope covers 3 or more variables– E.g. alldifferent(X1,X2,X3).– E.g. tour(X1,X2,X3,X4).

“Non-binary constraints” usually do not include unary constraints!

Constraint graph• Nodes = variables• Edge between 2 nodes

iff constraint between 2 associated variables– Few constraints, sparse

constraint graph– Lots of constraints,

dense constraint graph

Some non-binary examples

• Timetabling– Variables: Lecture1, Lecture2, …– Values: time1, time2, …– Constraint that lectures taught by same

lecturer do not conflict:

alldifferent(Lecture1,Lecture5,…).

Some non-binary examples

• Scheduling– Variables: Job1. Job2, …– Values: machine1, machine2, …– Constraint on number of jobs on each

machine:

atmost(2,[Job1,Job2,…],machine1),

atmost(1,[Job1,Job2,…],machine2).

Why use non-binary constraints?

• Binary constraints are NP-complete– Any non-binary constraint can be

represented using binary constraints– E.g. alldifferent(X1,X2,X3) is “equivalent”

to X1 =/= X2, X1 =/= X3, X2 =/= X3

• In theory therefore they’re not needed– But in practice, they are!

Modelling with non-binary constraints

• Benefits include:– Compact, declarative specifications

(discussed next)

– Efficient constraint propagation(discussed second)

Consider writing your own alldifferent constraint:

alldifferent([]).

alldifferent([Head|Tail]):-

onediff(Head,Tail),

alldifferent(Tail).

onediff(El,[]).

onediff(El,[Head|Tail]):-

El #\= Head,

onediff(El,Tail).

• It’s possible but it’s not very pleasant!

• Nor is it very compact– alldifferent([X1,…Xn]) expands into n(n-1)/2 binary not-

equals constraints, Xi \= Xj

– one non-binary constraint or O(n^2) binary constraints?

And there exist very efficient algorithms for reasoning efficiently with many specialized non-binary constraints

Constraint solvers• Two main approaches

– Systematic, tree search algorithms– Local search or repair based procedures

• Other more exotic possibilities– Hybrid algorithms– Quantum algorithms

Systematic solvers• Tree search

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

variables• Propagation to ensure some level of consistency

– If 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, …

Local search• Repair based methods

– Generate complete assignment– Change value to some variable in a violated

constraint

• Number of choices– Violated constraint, variable within it, …

Unable to exploit powerful constraint propagation techniques

Constraint propagation• Heart of constraint programming• Most often enforce 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

– A problem is AC iff every constraint is AC

Enforcing arc-consistency

• X2 \= X3 is AC• X1 \= X2 is not AC

– X2=1 has no support so can this value can be pruned

• X2 \= X3 is now not AC– No support for X3=2 – This value can also be

pruned

Problem is now AC

{1,2} {2,3}

Enforcing arc-consistency• Remove all values that are not AC

(i.e. have no support)

• May remove support from other values (often queue based algorithm)

• Best AC algorithms (AC7, AC-2000) run in O(ed^2)– Optimal if we know nothing else about the

constraints

Properties of AC• Unique maximal AC

subproblem– Or problem is

unsatisfiable

• Enforcing AC can process constraints in any order– But order does affect

(average-case) efficiency

Non-binary constraint propagation

• Most popular is generalized arc-consistency (GAC)– A non-binary constraint is GAC iff for every value

for a variable there are consistent values for all other variables in the constraint

– We can again prune values that are not supported

• GAC = AC on binary constraints

GAC on alldifferent• alldifferent(X1,X2,X3)

– Constraint is not GAC– X1=2 cannot be

extended• X2 would have to be 3• No value left then for

– X1={1} is GAC

{2,3}{2,3}

Enforcing GAC• Enforcing GAC is expensive in general

– GAC schema is O(d^k)On k-ary constraint on vars with domains of size d

• Trick is to exploit semantics of constraints– Regin’s all-different algorithm– Achieves GAC in just O(k^3/2 d)

On k-ary all different constraint with domains of size dBased on finding matching in “value graph”

Other types of constraint propagation

• (i,j)-consistency [due to Freuder, JACM 85]– Non-empty domains– Any consistent instantiation for i variables can be

extended to j others

• Describes many different consistency techniques

(i,j)-consistency • Generalization of arc-consistency

– AC = (1,1)-consistency– Path-consistency = (2,1)-consistency

• Strong path-consistency = AC + PC

– Path inverse consistency = (1,2)-consistency

Enforcing (i,j)-consistency• problem is (1,1)-consistent (AC)• BUT is not (2,1)-consistent (PC)

– X1=2, X2=3 cannot be extended to X3

– Need to add constraints:not(X1=2 & X2=3)

not(X1=2 & X3=3)

• Nor is it (1,2)-consistent (PIC)– X1=2 cannot be extended to X2 &

X3 (so needs to be deleted)

{2,3} {2,3}

Other types of constraint propagation

• Singleton arc-consistency (SAC)– Problem resulting from instantiating any variable

can be made AC

• Restricted path-consistency (RPC)– AC + if a value has just one support then any third

variable has a consistent value

• …

Comparing local consistencies

• Formal definition of tightness introduced by Debruyne & Bessiere [IJCAI-97]

• A-consistency is tighter than B-consistency iffIf a problem is A-consistent -> it is B-consistent

We write A >= B

Properties• Partial ordering

– reflexive A A– transitive A B & B C implies A C

• Defined relations– tighter A > B iff A B & not B A– incomparable A @ B iff neither A B nor B A

Comparison of consistency techniques

• Exercise for the reader, prove the following identities!

Strong PC > SAC > RPC > AC

NB gaps can reduce search exponentially!

Which to choose?• For binary constraints,

AC is often chosen– Space efficient

Just prune domains (cf PC)

– Time efficient

• For non-binary constraints GAC is often chosen– If we can exploit the

constraint semantics to keep it cheap!

Why consider these other consistencies?

• Promising experimental results– Useful pruning for their

additional cost

• Theoretical value– E.g. GAC on non-binary

constraints may exceed SAC on equivalent binary model

Maintaining a local consistency property

• Tree search– Assign value to variable– Enforce some level of local consistency

• Remove values/add new constraints

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

• Two popular algorithms– Maintaining arc-consistency (MAC)– Forward checking (only enforce AC on instantiated variable)

Modelling case study: Modelling case study: all interval seriesall interval series

Results due to Simonis, Puget & Results due to Simonis, Puget & ReginRegin

All interval series• Prob007 at www.csplib.org

• Comes from musical composition– Traced back to Alban Berg– Extensively used by Ernst Krenek

Op.170 “Quaestio temporis”

All interval series• Take the 12 standard pitch classes

– c, c#, d, ..– Represent them by numbers 0, .., 11

• Find a sequence so each occurs once– Each difference occurs once

All interval series• Can generalize to any n (not just 12)

Find Sn, a permutation of [0,n) such that |Sn+1-Sn| are all distinct

• Finding one solution is easy

All interval series• Can generalize to any n (not just 12)

Find Sn, a permutation of [0,n) such that |Sn+1-Sn| are all distinct

• Finding one solution is easy[n,1,n-1,2,n-2,.., floor(n/2)+2,floor(n/2)-1,floor(n/2)+1,floor(n/2)]

Giving the differences [n-1,n-2,..,2,1]

Challenge is to find all solutions!

Basic recipe• Devise basic CSP model

– What are the variables? What are the constraints?

• Introduce auxiliary variables if needed• Consider dual or combined models• Break symmetry• Introduce implied constraints

Basic CSP model• What are the variables?

Si = j if the ith note is j

• What are the constraints?

Basic CSP model• What are the variables?

Si = j if the ith note is j

• What are the constraints? Si in [0,n)

All-different([S1,S2,… Sn])

Forall i<i’ |Si+1 - Si| =/= |Si’+1 - Si’|

Improving basic model• Introduce auxiliary variables?

– Are there any loose or messy constraints we could better (more compactly?) express via some auxiliary variables?

Improving basic model• Introduce auxiliary variables?

– Yes, variables for the pairwise differences

Di = |Si+1 - Si|

• Now post single large all-different constraint

Di in [1,n-1]

All-different([D1,D2,…Dn-1])

Break symmetry• Does the problem have any symmetry?

– Yes, we can reverse any sequence

S1, S2, … Sn is an all-inverse series

Sn, …, S2, S1 is also

• How do we eliminate this symmetry?

– Yes, we can reverse any sequenceS1, S2, …, Sn is an all-inverse seriesSn, …, S2, S1 is also

• How do we eliminate this symmetry?• Order first and last difference

D1 < Dn-1

Break symmetry• Does the problem have any other

symmetry?

symmetry?– Yes, we can invert the numbers in any

sequence0, n-1, 1, n-2, … map x onto n-1-x

n-1, 0, n-2, 1, …

• How do we eliminate this symmetry?

symmetry?– Yes, we can invert the numbers in any

sequence0, n-1, 1, n-2, … map x onto n-1-xn-1, 0, n-2, 1, …

• How do we eliminate this symmetry?S1 < S2

Performance• Basic model is poor• Improved model able to compute all

solutions up to n=14 or so– GAC on all-different constraints very

beneficial– As is enforcing GAC on Di = |Si+1-Si|

This becomes too expensive for large nSo use just bounds consistency (BC) for larger n

Modelling case study: Modelling case study: Langford’s problemLangford’s problem

Model due to Barbara SmithModel due to Barbara Smith

Outline• Introduction

– Langford’s problem

• Modelling it as a CSP– Basic model– Refined model

• Experimental Results• Conclusions

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

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?

Variable for each occurrence of a numberX11 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,..X

– Of course, we can invert any sequence!

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)

What about dual model?• Can we take a dual view?

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

Dual model• What are the variables?

– Variable for each position i

• What are the values?

• 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

• 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

• 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)

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

Combined model• Primal and dual variables

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

Solving choices?• Which variables to assign?

– Xij or Di

– Xij or Di, doesn’t seem to matter

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

• 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

• 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

Global constraintsGlobal constraintsGlobal constraintsGlobal constraints

Non-exhaustive catalog• Order constraints• Constraints on values• Partitioning constraints• Timetabling constraints• Graph constraints• Scheduling constraints• Bin-packing constraints

Global constraints• It isn’t just all-different!

• Many constraints specialized to application domains– Scheduling– Packing– ..

Order constraints• min(X,[Y1,..,Yn]) and max(X,[Y1,..Yn])

X <= minimum(Y1,..,Yn)

X >= maximum(Y1,..Yn)

Order constraints• min_n(X,n,[Y1,..Ym]) and max_n(X,n,

[Y1,..,Ym)

X is nth smallest value in Y1,..YmX is nth largest value in Y1,..Ym

Value constraints• among(N,[Y1,..,Yn],[val1,..,valm])

N vars in [Y1,..,Yn] take values val1,..valme.g. among(2,[1,2,1,3,1,5],[3,4,5])

Value constraints• among(N,[Y1,..,Yn],[val1,..,valm])

N vars in [Y1,..,Yn] take values val1,..valme.g. among(2,[1,2,1,3,1,5],[3,4,5])

• count(n,[Y1,..,Ym],op,X) where op is =,<,>,=/,<= or >=relation “Yi op X” holds n times

among(n,[Y1,..,Ym],[k])=count(n,[Y1,..,Ym],=,k)

Value constraints• balance(N,[Y1,..,Yn])

N = occurrence of more frequent value - occurrence of least frequent value

E.g balance(2,[1,1,1,3,4,2])

Value constraints• balance(N,[Y1,..,Yn])

N = occurrence of more frequent value - occurrence of least frequent value

E.g balance(2,[1,1,1,3,4,2])

all-different([Y1,..,Yn]) => balance(0,[Y1,..,Yn])

Value constraints• min_nvalue(N,[Y1,..,Yn]) and max_nvalue(N,

[Y1,..,Yn])least (most) common value in Y1,..,Yn occurs N timesE.g. min_nvalue(2,[1,1,2,2,2,3,3,5,5])

Can replace multiple count or among constraints

Value constraints• common(X,Y,[X1,..,Xn],[Y1,..,Ym])

X vars in Xi take a value in Yi

Y vars in Yi take a value in Xi

E.g. common(3,4,[1,9,1,5],[2,1,9,9,6,9])

Value constraints• common(X,Y,[X1,..,Xn],[Y1,..,Ym])

X vars in Xi take a value in YiY vars in Yi take a value in XiE.g. common(3,4,[1,9,1,5],[2,1,9,9,6,9])

among(X,[Y1,..,Yn],[val1,..,valm]) = common(X,Y,[X1,..,Yn],[val1,..,valm])

Value constraints• same([X1,..,Xn],[Y1,..,Yn])

Yi is a permutation of Xi

• used_by([X1,..,Xn],[Y1,..,Ym])all values in Yi are used by vars in Xim>=m

• on n values:alldifferent([X1,..,Xn])=same([X1,..,Xn],[1,..,n])

=used_by([X1,..,Xn],[1,..,n])

Partitioning constraints• all-different([X1,..,Xn])

• Other flavoursall-different_except_0([X1,..,Xn])Xi=/Xj unless Xi=Xj=0

• 0 is often used for modelling purposes as “dummy” value

– Don’t use this slab– Don’t open this bin ..

Partitioning constraints• all-different([X1,..,Xn])

• Other flavourssymmetric-all-different([X1,..,Xn])Xi=/Xj and Xi=j iff Xj=i

• Very common in practice– Team i plays j iff Team j plays i..

Partitioning constraints• nvalue(N,[X1,..,Xn])

Xi takes N different valuesall-different([X1,..,Xn]) = nvalue(n,[X1,..,Xn)

Partitioning constraints• nvalue(N,[X1,..,Xn])

Xi takes N different valuesall-different([X1,..,Xn]) = nvalue(n,[X1,..,Xn)

• gcc([X1,..,Xn],Lo,Hi)values in Xi occur between Lo and Hi timesall-different([X1,..,Xn])=gcc([X1,..,Xn],1,1)

Timetabling constraints• change(N,[X1,..,Xn]),op) where op is

{=,<,>,<=,>=,/=}“Xi op Xi+1” holds N timesE.g. change(3,[4,4,3,4,1],/=)

• You may wish to limit the number of changes of classroom, shifts, …

Timetabling constraints• longest_changes(N,[X1,..,Xn]),op) where op

is {=,<,>,<=,>=,/=}longest sequence “Xi op Xi+1” is of length NE.g. longest_changes(2,[4,4,4,3,3,2,4,1,1,1],=)

• You may wish to limit the length of a shift without break, …

Graph constraints• Tours in graph a often represented by

the successors:

• [X1,..,Xn] means from node i we go to node Xi

Graph constraints• Tours in graph a often represented by

the successors:• [X1,..,Xn] means from node i we go to

node Xi• E.g. [2,1,5,3,4] represents the 2 cycles

(1)->(2)->(1) and (3)->(5)->(4)->(3)

Graph constraints• cycle(N,[X1,..,Xn])

there are N cycles in Xi

e.g. cycle(2,[2,1,5,3,4]) as we have the 2 cycles(1)->(2)->(1) and (3)->(5)->(4)->(3)

Useful for TSP like problems (e.g. sending engineers out to repair phones)

Scheduling constraints• cummulative([S1,..,Sn],[D1,..,Dn],[E1,..,En],

[H1,..,Hn],L)• schedules n (concurrent) jobs, each with a

height Hi• ith job starts at Si, runs for Di and ends at Ei

– Ei=Si+Di

• at any time, accumulated height of running jobs is less than L

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Scheduling constraints• coloured_cummulative([S1,..,Sn],[D1,..,Dn],

[E1,..,En],[C1,..,Cn],L)• schedules n (concurrent) each with a colour

Ci• no more than L colours running at any one

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Scheduling constraints• cycle_cummulative(m,[S1,..,Sn],

[D1,..,Dn],[E1,..,En],[H1,..,Hn],L)• schedules n (concurrent) jobs, each

with a height Hi onto a cyclic schedule of length m

Scheduling constraints• cummulatives([M1,,,Mn],[S1,..,Sn],[D1,..,Dn],

[E1,..,En],[H1,..,Hn],[L1,..,Lm])• schedules n (concurrent) jobs, each with a

height Hi onto one of m machines• ith runs on Mi• accumulated height of running jobs on

machine i <= Li

Scheduling constraints• cummulatives([M1,..Mn],[S1,..,Sn],

[D1,..,Dn],[E1,..,En],[H1,..,Hn],[L1,..,Lm])

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Machine 1

Machine 2

Scheduling constraints• coloured_cummulatives([M1,,,Mn],[S1,..,Sn],

[D1,..,Dn],[E1,..,En],[C1,..,Cn],[L1,..,Lm])• schedules n (concurrent) jobs, each with a

colour i onto one of m machines• ith runs on Mi• number of colours of running jobs on

machine i <= Li

Bin-packing constraints• bin_packing(capacity,[B1,..,Bn],

[w1,..,wn])

for each bin j, sum_Bi=j wi <= capacity

Bin-packing constraints• bin_packing(capacity,[B1,..,Bn],

[w1,..,wn])

for each bin j, sum_Bi=j wi <= capacity

special case of cummulative with task durations=1

Misc constraints• element(Index,[a1,..,an],Var)

Var=a_Index

constraint programming’s answer to arrays!

e.g. element(Item,[10,23,12,15],Cost)

Modelling: set variablesModelling: set variablesModelling: set variablesModelling: set variables

Motivation• Sets are useful when

– We don’t know how many objects we will have • e.g. set of items go in a bin

– We have symmetrical objects• e.g. items are symmetric and we don’t want to consider

all their permutations

And in many other situations!

Outline• Representing set variables

– Bounds– Characteristic functions

• Constraining set variables– Primitive constraints– Global constraints

Set variables• Representing sets

– Domain of values = powerset – Exponential space would be needed to represent this

extensional

– Domain of values = powerset – Exponential space would be needed to represent this

extensional– Compromise: just represent upper and lower bound

• E.g. {} subseteq X subseteq {1,2}

• X in {{},{1},{2},{1,2}}

– Domain of values = powerset

– Exponential space would be needed to represent this extensional

– Compromise: just represent upper and lower bound• E.g. {} subseteq X subseteq {1,2}• X in {{},{1},{2},{1,2}}

– Tradeoff• Cannot represent disjunction• E.g. X is {1} or {2} but not {1,2}

Alternative representation• Characteristic function

– i in X iff Xi=1

– i in X iff Xi=1– E.g. {1} subseteq X subseteq {1,2,3}

• X1=1• X2 in {0,1}, X3 in {0,1} • X4=X5=..=0

Primitive constraints• X subset Y• X subseteq Y• a in X• X = Y intersect Z• X = Y union Z• X = Y - Z• X = {}

Bounds consistency• Analogous to bounds consistency on

ordered finite domains

• Given constraint, C over X1,..,Xn

• C is BC if for each Xi, – a in glb(Xi) iff a is in some solution – a in lub(Xi) iff a is in all solutions

Bounds consistency• Lub(A union B) => Lub(A) union Lub(B)

• Glb(A union B) => Glb(A) union Glb(B)

• Glb(A union B) => Glb(A) union Glb(B)– This last rule is a safe approximation– Glb(A union B) superseteq Glb(A) union Glb(B)

• Lub(A intersect B) => Lub(A) intersect Lub(B)

• Glb(A intersect B) => Glb(A) intersect Glb(B)

• Glb(A intersect B) => Glb(A) intersect Glb(B)– The third rule is a safe approximation– Lub(A intersect B) subseteq

Lub(A) intersect Lub(B)

• A subseteq B => Lub(A) subseteq Lub(B),

Glb(A) subseteq Glb(B)

• A subseteq B => Lub(A) subseteq Lub(B),

Glb(A) subseteq Glb(B)

• A = B => A subseteq B, B subseteq A

Bounds consistency• Apply these rules exhaustively

• This computes safe approximations– Lub’s are correct or too large– Glb’s are correct or too small

• If a set of constraints have a solution– Exists unique lub and glb for the set variables– That satisfy these rules

Intervals v Characteristic functions

• It doesn’t really matter!

• TheoremA set of constraints in normal form is BC iff

the characteristic function representation is AC

Intervals v Characteristic functions

• It doesn’t really matter!

• TheoremA set of constraints in normal form is BC iff the

characteristic function representation is AC

NB characteristic function is 0/1 model so AC is the same as bounds consistency!

Global constraints• All-different constraint

– No two vars take same value

• Permutation constraint– Special case of all-different– All values are used

• Disjoint constraint– No two set variables intersect

• Permutation constraint– All-different, & all values are used

• Partition constraint– Disjoint, & all values are used

Global constraints• Disjoint([X1,..,Xn]) decomposes into

Xi intersect Xj = {}

Just like the all-different constraint!

But decomposition does not hurt!

Theorem

BC on disjoint([X1,..,Xn]) =

BC on decomposed model

Global constraints• Partition([X1,..,Xn],S) decomposes into

Xi intesect Xj = {}

X1 union X2=Y1

X3 union Y1 = Y2

Xn union Yn-2 = S

Global constraints• Partition([X1,..,Xn],S) decomposes into

Xi intesect Xj = {}

X1 union X2=Y1

X3 union Y1 = Y2

Xn union Yn-2 = S

Decomposition does not hurt!

Global constraints• Partition([X1,..,Xn],S) decomposes

Decomposition again does not hurt!

TheoremBC on partition([X1,..,Xn],S) =BC on the decomposed model

Global constraints• Non-empty-partition([X1,..,Xn],S)

• Decomposes as before

• With additional constraint:– |Xi| > 0

Decomposition now hurts!

Decomposition now hurts!TheoremBC on non-empty-partition([X1,..,Xn],S) is strictly stronger than BC on decompostion

Modelling with set constraints

• Ternary Steiner problem– n(n-1)/6 sets, Si – Each a subset of {1,..,n}– |Si| = 3

Nb n(n-1)/6 = nC2 / 3

Modelling with set constraints

• Ternary Steiner problem– n(n-1)/6 sets, Si – Each a subset of {1,..,n}– |Si| = 3– |Si intersect Sj| <=1

Ternary Steiner problem• Only soluble if n mod 6 = 1 or 3 [Kirkman 1847]

– S3 = {{1,2,3}}– S7 = {{1,2,4},{2,3,5},{3,4,6},{4,5,7},{5,6,1},{6,7,2},{7,1,3}}

• Number of non-isomorphic triples– 1,1,2,80,>1.1*10^9 [Colbourn & Dinitz 1996]– AO30139

Ternary Steiner problem• Simple and elegant set variable model

Let t=n(n-1)/6For all j in [1,t] . Sj subset {1,..,n} and |Sj|=3Atmost1([S1,..,St])

For all j<k in [1,t] . |Sj intersect Sk| <=1

What’s wrong with this model?

What’s wrong with this model?Numbers and Si are still symmetric.How do we deal with this?

Ternary Steiner problem– Breaking symmetry

• Make some initial assignments• S1={1,2,3}, S2={2,4,5}, …• Until all numbers named once

Ternary Steiner problem– Breaking symmetry

• Order sets S1<S2<..<St • e.g. multiset ordering

Conclusions• Set variables are useful for modelling

– But they still can leave symmetry!

• Constraints on set variables– Primitive constraints– Global constraints

• Decomposition hurts in general as soon as we add cardinality conditions

– Level of consistency

Solving choices• Level of consistency

– Binary v non-binary constraints– GAC, BC …

Solving choices• Branching heuristics

– Variable ordering (fail first)• Smallest domain, max degree, …

– Value ordering (succed first)• Max promise, …

– Level of consistency– Variable, value ordering

Modelling & Solving with Constraints Prof. Toby Walsh University College Cork/Uppsala University.

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