Non-binary constraints: modelling

Toby Walsh

Cork Constraint Computation Center

Modelling case studies

• Golomb rulers Graceful graphs [Smith & Walsh

AAAI2000]

• Permutation problems Channelling

constraints [Walsh LPAR 2001]

Golomb rulers

• Mark ticks on a ruler Distance between any two ticks (not just

neighbouring ticks) is distinct

• Applications in radio-astronomy, cystallography, … http://www.csplib.org/prob/prob006

Golomb rulers

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

• Goal is to find minimal 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 rulers

0,1

0,1,3

0,1,4,6

0,1,4,9,11

0,1,4,10,12,17

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

Solutions grow as approximately O(n^2)

Modelling the Golomb ruler

• Variable, Xi for each tick

• Value is position on ruler

• Naïve model with quaternary constraints For all i>j,k>l>j |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,…]). Tighter constraints and denser constraint graph

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 modelling issues

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

E.g. D12 < D13, D23 < D24, …

E.g. D1k at least sum of first k integers

• 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 order Heuristics like fail-first (smallest domain) not

effective on this problem!

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

Model (re)formulation

• Can we automate the process of refining and improving a model? Identify and breaking

symmetries Inferring implied

constraints …

CGRASS proof planner hopes to do this!

The `Introduce` Method

Preconditions1. Exp occurs more than once in the constraint set.2. someVariable = Exp not already present.Post-conditions1. Generate new var, x, domain defined by Exp.2. Add constraint: x = Exp.

Motivation: solver propagates through newly introduced variable

Other automatic methods

• Symmetry identification and removal Order ticks

• Variable elimination Generalization of Gaussian

elimination

• Automatically turns naïve Golomb model into a reasonable one!

Something to try at home?

• Circular (or modular) Golomb rulers Inter-tick distance

variables more central, removing rotational symmetry?

• 2-d Golomb rulers

All examples of “graceful” graphs

Summary

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

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

• Large space of models Non-binary constraints only make this worse? Automatic tools can help with model selection and

reformulation

Summary (II)

• Modelling decisions: Auxiliary variables Implied constraints Symmetry breaking constraints

• More to constraints than just declarative problem specifications!

Case study II: permutation problems

Wide variety of scheduling, assignment and routing problems involve finding a permutation Plus satisfying some

additional constraints

Illustrates some of the fundamental decisions that must be made when modelling!

Modelling choices

• Need to decide variables, domains and constraints

• Often difficult choice for something even as basic as the decision variables?

E.g. consider scheduling the World Cup.

Are vars=games, vals=times

or vars=times, vals=games ?

Permutation problems

• |vars|=|vals| each var has unique val

• many examples scheduling timetabling routing assignment problems

• permute vars for vals which do we choose?TSP problem = find permutation of cities

which makes a tour of minimum length

Meta-motivation

• Methodology for comparing models based on definition of

constraint tightness

• Other applications comparing implied constraints impact of reformulation

Comparing models

• Choice of model affects amount of constraint propagation tighter model more pruning and propagation

• Need dynamic tightness measure domains shrink as we descend down search tree other constraints may prune domains

Constraint tightness

• Pruning depends on local consistency enforced higher consistencies will infer implied constraints

missing from looser models

• Introduce tightness measure: parameterized by level of consistency enforced considers domains changing in size

Definition of tightness

• model 1 is as tight as model 2 wrt A-consistency iff

given any domains

model 1 is A-consistent -> model 2 is A-consistent

• written A1 A2

How does this compare to the previous definition?

• Tightness measure introduced by Debruyne & Bessiere [IJCAI-97]

• A-consistency is tighter than B-consistency iff given any modelthe model is A-consistent -> the model is B-consistent

• We fix models but vary domains

Properties of new ordering

• Partial ordering reflexive A1 A1 transitive A1 A2 & A2 A3 implies A1 A3

• Defined relations tighter A1 A2 iff A1 A2 & not A2 A1

equivalence A1 = A2 iff A1 A2 & A2 A1

incomparable A1 @ A2 iff neither A1 A2 nor A2 A1

Further properties

• Monotonicity AC1u2 AC1 AC1n2

adding constraints can only tighten a model

• Fixed point AC1 AC2 implies AC1u2 = AC1

combining a looser model with a tighter model doesn’t help!

Extensions

• Ordering extends to search algorithms E.g. MAC1 MAC2 iff, given any domains, MAC on

model 1 visits no more nodes than MAC on model 2 assume equivalent var and val ordering

Similar monotonicity and fixed point properties

• Ordering extends to different consistencies applied to the different models E.g. GAC1 AC2

Permutation models

• primal model xi xj for all i,j

• primal alldiff model alldifferent(x1,x2,…)

• primal/dual models (dual) variables

associated with each (primal) value

“Modelling a Permutation Problem”, Barbara M Smith, ECAI'2000 Workshop on Modelling and Solving Problems with Constraints

Primal model

• n primal variables each with n values

• O(n^2) binary constraints

xi xj

x5

x6

x1 x2

x3

x4

Primal all-different model

• n primal variables each with n values

• one non-binary constraint

all-different(x1,x2,..)x6

x5 x4

x3

x2x1

Primal/dual model

• n primal variables each with n values

• n dual variables one for each primal value each dual value associated

with a primal variable

• n^2 channelling constraints

xi=j iff dj=i

• no other constraints needed!

x5

x1

x2

x3

x4

d5

d1

d2

d3

d4

Other reason to channel

• Channelling between models frequent modelling technique

• Some constraints easier to specify in one model

• Others easier to specify in a (dual) model

• Channelling maintains consistency between the 2 models

Multiple permutation problems

• Some problems consist of several permutations order n quasigroup (or

Latin square) has 2n intersecting permuations each of size n

• Following results extend to such cases

Some notation

BC bounds consistency AC arc consistency RPC restricted path consistency PIC path inverse consistency SAC singleton arc consistency ACPC strong path consistency GAC generalized arc consistency

↓increasingpruning

Some notation

primal not equals primal alldiff c channelling constraints c channelling constraints & primal not equals c channelling constraints & primal alldiff c channelling constraints & primal/dual not equals...

Theoretical results

SAT models

• n Boolean vars, Xij true iff xi=j• primal SAT model

O(n) clauses, each var takes at least one val O(n^3) clauses, no primal var takes two vals O(n^3) clauses, no two primal vars take same val

• channelling SAT model no need for (dual) Boolean vars as Xij can be used again

[Gomes et al 2001] and [Bejar & Manya 2000] report promising experimental results using SAT models of permutation problems with the Davis Putnam (DP) tree search algorithm

Theoretical results

MAC tighter than DP, DP as tight as FC

Asymptotic results

• Tightness ordering reflects asymptotic cost E.g. GAC ACc AC

O(n^4) O(n^3) O(n^2)

In each case, using best known algorithm

Hence we need to run experiments to know if extra pruning is worth the cost!

Experimental results

• [Smith 2000] reports promising results for MACc on Langford’s problem

• I studied 3 other permutation problems all interval series circular Golomb rulers quasigroups (Latin squares)

All 4 problems in www.csplib.org

Experimental results

• using Sicstus FD constraint library channelling is a standard non-binary constraint

• MACc best on “looser” problems Langford’s problem, all interval series

• Maintaining GAC better on “tighter” problems circular Golomb rulers, mean (but not median)

quasigroup performance

Extensions

• injective mappings more vals than vars, each var takes unique val can introduce dummy vars to make permutation

• channelling constraints useful in many other problems often (but not always) bijective propagate pruning rapidly between models

Conclusions

• Many ways to model and solve even something as simple as a permutation problem Hard even to decide what are the variables!

• Theory and experiment needed to compare models properly Dominance/asymptotic results only take us so far

• Best model may not be a single model! Channel between two (or more?) models