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Constraint Programming

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Title: Constraint Programming

1
Constraint Programming

Toby Walsh
UNSW and NICTA

2
Overview

Constraint satisfaction
What is a constraint satisfaction problem? (aka
CSP)
Constraint propagation (aka inference)
Search algorithms for solving CSPs
Modelling problems using CSPs
Some case studies
Auxiliary variables
Symmetry
Implied constraints
Dual models

3
Resources

Course links
http//www.cse.unsw.edu.au/tw/cp.html
Benchmark problems
www.csplib.org
Constraints solvers
Logic programming languages ECLIPSE, BProlog
Functional languages FaCiLe (OCaml),
Imperative languages Choco (Java),

4
Constraint programming

Ongoing dream of declarative programming
State the constraints
No two exams at the same time
Only 5 shifts in one week
Parcel picked up between 10am-noon
Black box solver finds a solution
Just like magic!
Paradigm of choice for many hard combinatorial
problems
Scheduling, assignment, routing,

5
Constraints are everywhere!

No meetings before 10am
Network traffic lt 100 Gbytes/sec
PCB width lt 21cm
Salary gt 45k Euros
No two exams at the same time
Only 5 shifts in one week
Parcel picked up before 10am

6
Constraint satisfaction

Constraint satisfaction problem (CSP) consists
of
Set of variables
Each variable has set of values
Usually assume finite domain
true,false, red,blue,green, 0,10,
Set of constraints
Goal
Find assignment of values to variables to satisfy
all the constraints

7
Example CSP

Sudoku
Variable for each square
Domain numbers 1 to 9
Constraints
Row
AllDifferent(X1,..X9)
AllDifferent(X10,..X18)
Column
AllDifferent(X1,X10,..)
Small square
AllDifferent(X1,..,X3,X10,..X12,..)

8
Example CSP

Course timetabling
Variable for each course
X1, X2 ..
Domain are possible times for course
Wed9am, Fri10am, ..
Constraints
X1 \ Wed9am
Capacity constraints atmost(3,X1,X2..,Wed9am)
Lecturer constraints alldifferent(X1,X5,)

9
Constraint optimization

CSP objective function
E.g. objective is Profit Income - Costs
Find assignment of values to variables that
Satisfies constraints
Maximizes (minimizes) objective
Often solved as sequence of satisfaction problems
Profit gt 0, Profit gt Ans1, Profit gt Ans2,

10
Constraint programming v. 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,

11
Constraints

Constraints
Scope
list of variables to which constraint applies
Relation specifying allowed values (goods)
Extensionally specified
(Xred, Yblue) or (Xred, Ygreen) or
(Xblue, Yred) or (Xblue, Ygreen) or
(Xgreen, Yred) or (Xgreen, Yblue)
Intensionally specified
X / Y
X 2Y - Z lt 5
Alldifferent(X,Y,Z)
AtLeast(1,X,Y,Z,red)

12
Binary v non-binary

Binary constraint
Scope covers 2 variables
E.g. not-equals constraint X1 / X2.
E.g. ordering constraint X1 lt 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!

13
Some non-binary examples

Timetabling
Variables Lecture1, Lecture2,
Values time1, time2,
Constraint that lectures taught by same lecturer
do not conflict
alldifferent(Lecture1,Lecture5,).

14
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).

15
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 theyre not needed
But in practice, they are!

16
Modelling with non-binary constraints

Benefits include
Compact, declarative specifications
(discussed next)
Efficient constraint propagation
(discussed second)

17
Modelling with non-binary constraints

Consider writing your own alldifferent
constraint
alldifferent().
alldifferent(HeadTail)-
onediff(Head,Tail),
alldifferent(Tail).

onediff(El,).
onediff(El,HeadTail)-
El \ Head,
onediff(El,Tail).

18
Constraint solvers

Two main approaches
Systematic, tree search algorithms
Local search or repair based procedures
Other more exotic possibilities
Hybrid algorithms
Quantum algorithms

19
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,

20
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

21
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

22
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(ed2)
Optimal if we know nothing else about the
constraints

23
Enforcing arc-consistency

Consider 1st column
Binary not equals constraints
X1/X10, X1/X19, ..
Consider 8th row
Look at domain of the 1st variable

24
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

25
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

26
GAC on alldifferent

AllDifferent on 6th row
1,2,4,5,6,7
1,2,4,5,6,7
8
9
1,2,4,5,6,7
1,2,4,5,6,7
1,2,4,5,6,7
3
1,2,4,5,6,7

27
GAC on alldifferent

AllDifferent on 1st col
6,7
1,2,4,5,6,7
8
9
1,2,4,5,6,7
1,2,4,5,6,7
1,2,4,5,6,7
3
1,2,4,5,6,7

28
GAC on alldifferent

AllDifferent on 2nd col
6,7
1,2,4,5
8
9
1,2,4,5,6,7
1,2,4,5,6,7
1,2,4,5,6,7
3
1,2,4,5,6,7

29
GAC on alldifferent

AllDifferent on 4th small square
6,7
1,2,4
8
9
1,2,4,5,6,7
1,2,4,5,6,7
1,2,4,5,6,7
3
1,2,4,5,6,7

30
GAC on alldifferent

AllDifferent on 5th col
6,7
1,2,4
8
9
1,2,5,6,7
1,2,4,5,6,7
1,2,4,5,6,7
3
1,2,4,5,6,7

31
GAC on alldifferent

AllDifferent on 6th col
6,7
1,2,4
8
9
1,2,5,6,7
1,4,5,7
1,2,4,5,6,7
3
1,2,4,5,6,7

32
GAC on alldifferent

AllDifferent on 5th small square
6,7
1,2,4
8
9
5,7
4,5
1,2,4,5,6,7
3
1,2,4,5,6,7

33
GAC on alldifferent

AllDifferent on 7th col
6,7
1,2,4
8
9
5,7
4,5
1,7
3
1,2,4,5,6,7

34
GAC on alldifferent

AllDifferent on 9th col
6,7
1,2,4
8
9
5,7
4,5
1,7
3
4,5,6

35
GAC on alldifferent

AllDifferent on 6th small square
6,7
1,2,4
8
9
5,7
4,5
1,7
3
4,6

36
Enforcing GAC

Enforcing GAC is expensive in general
GAC schema is O(dk)
On k-ary constraint on vars with domains of size
d
Trick is to exploit semantics of constraints
Regins all-different algorithm
Achieves GAC in just O(k2 d2)
On k-ary all different constraint with domains of
size d
Based on finding matching in value graph

37
Other types of constraint propagation

(i,j)-consistency
Non-empty domains
Any consistent instantiation for i variables can
be extended to j others
Describes many different consistency techniques

38
(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

39
Enforcing (i,j)-consistency

problem is (1,1)-consistent (AC)
BUT is not (2,1)-consistent (PC)
X12, X23 cannot be extended to X3
Need to add constraints
not(X12 X23)
not(X12 X33)
Nor is it (1,2)-consistent (PIC)
X12 cannot be extended to X2 X3 (so needs to
be deleted)

1,2
X1
\
\
2,3
2,3
\
X3
X2
40
Other types of constraint propagation

Bounds consistency (BC)
With ordered domains
Enforce AC just on max/min elements
Used often for arithmetic constraints (eg linear
inequalities)
GAC is intractable on a sum constraints ( subset
sum)

41
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)
Make the whole problem arc-consistent
Forward checking (very restricted form of AC
maintained)

42
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

43
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

44
n-queens problem