Non-binary Constraints

Toby Walshtw@cs.york.ac.uk

Outline

Definition of non-binary constraints Modeling with non-binary constraints Constraint propagation with non-binary

constraints Practical benefits: case study

– Golomb rulers

Definitions

Binary constraint– Relation on 2 variables identifying those pairs

of values disallowed (nogoods)– E.g. not-equals constraint: X1 \= X2.

Non-binary constraint– Relation on 3 or more variables identifying

tuples of values disallowed– E.g. alldifferent(X1,X2,X3).

Some non-binary examples

Timetabling– Variables: Lecture1. Lecture2, …– Values: time1, time2, …– Constraint that lectures do not conflict:

alldifferent(Lecture1,Lecture2,…).

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!

Modeling with non-binary constraints

Benefits include:– Compact, declarative specifications

(discussed next)

– Efficient constraint propagation(discussed after next section)

Modeling with non-binary constraints

Consider writing your own alldifferent constraint:

alldifferent([]).alldifferent([Head|Tail]):-

onedifferent(Head,Tail),alldifferent(Tail).

onedifferent(El,[]).onedifferent(El,[Head|Tail]):-

El \= Head,onedifferent(El,Tail).

Modeling with non-binary constraints

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?

Theoretical comparison

Constraint algorithms:– Tree search (labeling)– Constraint propagation at each node

Binary constraint propagation– Arc-consistency

Non-binary constraint propagation– Generalized arc-consistency

Binary constraint propagation

Arc-consistency (AC) is very popular– 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

– We can prune values that are not supported – A problem is AC iff every constraint is AC

AC offers good tradeoff between amount of pruning and computational effort

Binary constraint propagation

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}

{1,2} {2,3}

\=

\=

X1

X3X2

Non-binary constraint propagation

generalized arc-consistency (GAC) for non-binary constraints– 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 prune values that are not supported

GAC = AC on binary constraints

GAC is stronger than AC

Pigeonhole problem– 3 pigeons in 2 holes

Non-binary model– alldifferent(X1,X2,X3)

is not GAC

Binary model– X1 \= X2, X1 \= X3,

X2 \= X3 are all AC

{2,3}

{2,3} {2,3}

X1

X3X2

\= \=

\=

Using GAC within search

Tree search– Instantiate chosen variable with value (label)– Maintain (incrementally enfoce) some level of

consistency Maintaining GAC can be exponentially

better than maintaining AC– Construct generalized pigeonhole example on

which we’ll explore exponentially fewer nodes

Achieving GAC

By exploiting “semantics” of constraints, we can often enforce GAC efficiently– Consider alldifferent([X1,…Xn]) with each Xi

having domain of size m

– Generic GAC algorithm runs in O(m^n)

– Specialized GAC algorithm for alldifferent runs in O(m^2 n^2) based on network flow

Practical benefits

How do these (theoretical) differences affect you practically?

Case study– Golomb rulers

Golomb rulers

Mark ticks on a ruler– Distance between any two (not necessarily

consecutive) ticks is distinct

Applications in radio-astronomy– http://csplib.cs.strath.ac.uk/prob006

Golomb rulers

Simple solution– Exponentially long ruler– Ticks at 0,1,3,7,15,31,63,…

Goal is to find miminal length rulers– turn optimization problem into sequence of

satisfaction problemsIs there a ruler of length m?Is there a ruler of length m-1?….

Optimal Golomb rulers

Known for up to 23 ticks Distributed internet project to find large

rulers0,1

0,1,30,1,4,6

0,1,4,9,110,1,4,10,12,17

0,1,4,10,18,23,25

Modeling the Golomb ruler

Variable, Xi for each tick Value is position on ruler

Naïve model with quaternary constraints– For all i,j,k,l |Xi-Xj| \= |Xk-Xl|

Problems with naïve model

Large number of quaternary constraints– O(n^4) constraints

Looseness of quaternary constraints– Many values satisfy |Xi-Xj| \= |Xk-Xl|– Limited pruning

A better non-binary model

Introduce auxiliary variables for inter-tick distances– Dij = |Xi-Xj|– O(n^2) ternary constraints

Post single large non-binary constraint– alldifferent([D11,D12,…]).– Relatively tight!

Other modeling issues

Symmetry– A ruler can always be reversed!– Break this symmetry by adding constraint:

D12 < Dn-1,n– Also break symmetry on Xi

X1 < X2 < … Xn– Such tricks important in many problems

Other modeling issues

Additional (implied) constraints– Don’t change set of solutions– But may reduce search significantly

E.g. D12 < D13, D23 < D24, … Pure declarative specifications are not

enough!

Solving issues

Labeling strategies often very important– Smallest domain often good idea– Focuses on “hardest” part of problem

Best strategy for Golomb ruler is instantiate variables in strict

Experimental results

Runtime/sec

Naïve model

Alldifferent model

8-Find 2.0 0.1

8-Prove 12.0 10.2

9-Find 31.7 1.6

9-Prove 168 9.7

10-Find 657 24.3

10-Prove > 10^5 68.3

Something to try at home?

Circular (or modular) Golomb rulers

2-d Golomb rulers

Conclusions

Benefits of non-binary constraints– Compact, declarative models– Efficient and effective constraint propagation

Supported by many constraint toolkits– alldifferent, atmost, cardinality, …

Conclusions

Modeling decisions:– Auxiliary variables– Implied constraints– Symmetry breaking constraints

More to constraints than just declarative problem specifications!