Non-Binary Constraint Satisfaction Toby Walsh Cork Constraint Computation Center.
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Transcript of Non-Binary Constraint Satisfaction Toby Walsh Cork Constraint Computation Center.
Non-Binary Constraint Non-Binary Constraint SatisfactionSatisfaction
Non-Binary Constraint Non-Binary Constraint SatisfactionSatisfaction
Toby WalshToby Walsh
Cork Constraint Computation CenterCork Constraint Computation Center
Overview• Introduction to CSPs
– Local Consistencies, search algorithms
• Modelling (4 case studies)– Auxiliary variables, implied constraints– Redundant models and channelling constraints
• Eliminating non-binary constraints– Encodings, decomposition
• Theoretical properties– Tightness, relational consistencies
Resources• Course links
– www.cs.york.ac.uk/~tw/Links/csps/
• Benchmark problems– www.csplib.org
• Constraints solvers– LP based like ECLIPSE, Java based liked
Jsolver, open source like Choco, …
Constraint programming• Ongoing “dream” of declarative
programming– State the constraints– Solver finds a solution
• Paradigm 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 courseX1, X2 ..
– Domain are possible times for course
Wed9am, Fri10am, ..
– Constraints:X1 \= Wed9amCapacity constraints: atmost(3,
[X1,X2..],Wed9am)Lecturer constraints:
alldifferent([X1,X5,…])
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)
• But can also define constraint satisfaction or optimization within an imperative of functional language• Popular toolkits in 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]), …
Constraint tightness• Informally, tight
constraints admit few tuples– E.g. on 0/1 vars
X1 =/= X2 is loose as half tuples satisfy
X1+X2+X3+X4+X5 <= 1 is tight as only 5 out of 32 tuples satisfy
• More formal definition later
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)
Modelling with non-binary constraints
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).
Modelling 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?
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• 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• 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
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}
{1,2} {2,3}
\=
\=
X1
X3X2
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
X3
– X1={1} is GAC
{1,2}
{2,3}{2,3}
X1
X2 X3
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^2 d^2)
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)
{1,2}
{2,3} {2,3}
\=
\=
X1
X3X2
\=
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
Other types of consistency• problem is (1,1)-consistent (AC)• BUT is not singleton AC (SAC)
– Problem with X1=2 cannot be made AC (so value should be deleted)
• Nor is it restricted PC (RPC)– X1=2 has only one support in X2
(the value 3) but X3 then has no consistent values
– X1=2 can therefore be deleted
{1,2}
{2,3} {2,3}
\=
\=
X1
X3X2
\=
Other types of constraint propagation
• Neighbourhood inverse consistency (NIC)– For all vals for a var, there are consistent vals for all vars in
the immediate neighbourhood
• Bounds consistency (BC)– With ordered domains– Enforce AC just on max/min elements
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 > PIC > RPC > AC > BC NIC > PICNIC @ SACNIC @ Strong PC
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 (very restricted form of AC maintained)
Forward checking• Binary constraints (FC)
– Make constraints involving current variable and one future variable arc-consistent
– No need to look at any other constraints!
• Non-binary constraints– Several choices as to how to do forward
checking
Forward checking with non-binary constraints
nFC0 makes AC only those k-ary constraints with k-1 variables set
nFC1 applies one pass of AC on constraints and projections involving current var and one future var
nFC2 applies one pass of GAC on constraints involving current var and at least one future var
nFC3 enforces GAC on this set
nFC4 applies one pass of GAC on constraints involving at least one past and one future var
nFC5 enforces GAC on this set
Summary• Constraint solving
– Constraint propagation central part of many algorithms
• Binary v non-binary constraints– Compact, more
efficient representation