Finite Constraint Domains

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Finite Constraint Domains

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Finite Constraint Domains

• Constraint satisfaction problems (CSP)
• A backtracking solver
• Node and arc consistency
• Bounds consistency
• Generalized consistency
• Optimization for arithmetic CSPs

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Finite Constraint Domains

• An important class of constraint domains
• Use to model constraint problems involving
  choice e.G. Scheduling, routing and timetabling
• The greatest industrial impact of constraint
  programming has been on these problems

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Constraint Satisfaction Problems

• A constraint satisfaction problem (CSP) consists
  of
• A constraint C over variables x1,..., Xn
• A domain D which maps each variable xi to a set
  of possible values d(xi)
• It is understood as the constraint

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Map Colouring
A classic CSP is the problem of coloring a map so
that no adjacent regions have the same color
Can the map of Australia be colored with 3 colors
?
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4-queens
Place 4 queens on a 4 x 4 chessboard so that none
can take another.
Four variables Q1, Q2, Q3, Q4 representing the
row of the queen in each column. Domain of each
variable is 1,2,3,4
One solution! --gt
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4-queens
The constraints
Not on the same row Not diagonally up Not
diagonally down
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Smugglers Knapsack
Smuggler with knapsack with capacity 9, who needs
to choose items to smuggle to make profit at
least 30
What should be the domains of the variables?
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Systematic Search Methods

• exploring the solution space
• complete and sound
• efficiency issues
• Backtracking (BT)
• Generate Test (GT)

exploring subspace
exploringindividual assignments
Z
Y
X
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Generate Test

• probably the most general problem solving method
• Algorithm
• generate labelling
• test satisfaction
• Drawbacks Improvements
• blind generator smart generator --gt local
  search
• late discovery of testing within generator
• inconsistencies --gt backtracking

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Backtracking (BT)

• incrementally extends a partial solution towards
  a complete solution
• Algorithm
• assign value to variable
• check consistency
• until all variables labelled
• Drawbacks
• thrashing
• redundant work
• late detection of conflict

A
1
B
2
1
C
2
1
1
D
2
2
1
1
1
?
A D, B ? D, AC lt 4
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GT BT - Example

• Problem X1,2, Y1,2, Z1,2 X Y, X
  ? Z, Y gt Z
• generate test backtracking

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Simple Backtracking Solver

• The simplest way to solve CSPs is to enumerate
  the possible solutions
• The backtracking solver
• Enumerates values for one variable at a time
• Checks that no prim. Constraint is false at each
  stage
• Assume satisfiable(c) returns false when
  primitive constraint c with no variables is
  unsatisfiable

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Partial Satisfiable

• Check whether a constraint is unsatisfiable
  because of a prim. Constraint with no vars
• Partial_satisfiable(c)
• For each primitive constraint c in C
• If vars(c) is empty
• If satisfiable(c) false return false
• Return true

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Backtrack Solve

• Back_solve(c,d)
• If vars(c) is empty return partial_satisfiable(c)
  
• Choose x in vars(c)
• For each value d in d(x)
• Let C1 be C with x replaced by d
• If partial_satisfiable(c1) then
• If back_solve(c1,d) then return true
• Return false

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Backtracking Solve
Choose var X domain 1,2
Choose var Y domain 1,2
Choose var Z domain 1,2
Variable X domain 1,2
Choose var Y domain 1,2
No variables, and false
partial_satisfiable false
No variables, and false
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Consistency Techniques

• removing inconsistent values from variables
  domains
• graph representation of the CSP
• binary and unary constraints only (no problem!)
• nodes variables
• edges constraints
• node consistency (NC)
• arc consistency (AC)
• path consistency (PC)
• (strong) k-consistency

Agt5
A
AltC
A?B
C
B
BC
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Node and Arc Consistency

• Basic idea find an equivalent CSP to the
  original one with smaller domains of vars
• Key examine 1 prim.Constraint c at a time
• Node consistency (vars(c)x) remove any values
  from domain of x that falsify c
• Arc consistency (vars(c)x,y) remove any
  values from d(x) for which there is no value in
  d(y) that satisfies c and vice versa

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Node Consistency

• Primitive constraint c is node consistent with
  domain D if vars(c) /1 or
• If vars(c) x then for each d in d(x)
• X assigned d is a solution of c
• A CSP is node consistent if each prim. Constraint
  in it is node consistent

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Node Consistency Examples
Example CSP is not node consistent (see Z)
This CSP is node consistent
The map coloring and 4-queens CSPs are node
consistent. Why?
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Achieving Node Consistency

• Node_consistent(c,d)
• For each prim. Constraint c in C
• D node_consistent_primitive(c, D)
• Return D
• Node_consistent_primitive(c, D)
• If vars(c) 1 then
• Let x vars(c)
• Return D

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Arc Consistency

• A primitive constraint c is arc consistent with
  domain D if varsc ! 2 or
• Vars(c) x,y and for each d in d(x) there
  exists e in d(y) such that
• And similarly for y
• A CSP is arc consistent if each prim. Constraint
  in it is arc consistent

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Arc Consistency Examples
This CSP is node consistent but not arc
consistent
For example the value 4 for X and X lt Y. The
following equivalent CSP is arc consistent
The map coloring and 4-queens CSPs are also arc
consistent.
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Achieving Arc Consistency

• Arc_consistent_primitive(c, D)
• If vars(c) 2 then
• Return D
• Removes values which are not arc consistent with c

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Achieving Arc Consistency

• Arc_consistent(c,d)
• Repeat
• W d
• For each prim. Constraint c in C
• D arc_consistent_primitive(c,d)
• Until W D
• Return D
• A very naive version (there are much better)

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Using Node and Arc Cons.

• We can build constraint solvers using the
  consistency methods
• Two important kinds of domain
• False domain some variable has empty domain
• Valuation domain each variable has a singleton
  domain
• Extend satisfiable to CSP with val. Domain

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Node and Arc Cons. Solver

• D node_consistent(C,D)
• D arc_consistent(C,D)
• if D is a false domain
• return false
• if D is a valuation domain
• return satisfiable(C,D)
• return unknown

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Node and Arc Solver Example
Colouring Australia with constraints
Node consistency
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Node and Arc Solver Example
Colouring Australia with constraints
Arc consistency
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Node and Arc Solver Example
Colouring Australia with constraints
Arc consistency
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Node and Arc Solver Example
Colouring Australia with constraints
Arc consistency
Answer unknown
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Backtracking Cons. Solver

• We can combine consistency with the backtracking
  solver
• Apply node and arc consistency before starting
  the backtracking solver and after each variable
  is given a value

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Back. Cons Solver Example
Q1
Q2
Q3
Q4
1
No value can be assigned to Q3 in this case!
Therefore, we need to choose another value for
Q2.
There is no possible value for variable Q3!
2
3
4
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Back. Cons Solver Example
Q1
Q2
Q3
Q4
1
Backtracking Find another value for Q3? No!
backtracking, Find another value of Q2? No!
backtracking, Find another value of Q1? Yes, Q1
2
We cannot find any possible value for Q4
in this case!
2
3
4
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Back. Cons Solver Example
Q1
Q2
Q3
Q4
1
2
3
4
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Back. Cons Solver Example
Q1
Q2
Q3
Q4
1
2
3
4
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Node and Arc Solver Example
Colouring Australia with constraints
Backtracking enumeration
Select a variable with domain of more than 1, T
Add constraint
Apply consistency
Answer true
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Is AC enough?

• empty domain gt no solution
• cardinality of all domains is 1 gt solution
• Problem X1,2, Y1,2, Z1,2 X ? Y, X
  ? Z, Y ? Z

X
1 2
1 2
Y
Z
1 2
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Path Consistency (PC)
V2
V4
V3
V5
V0
V1
???

• Path (V0 Vn) is path consistent iff for each
  pair of compatible values x in D(V0) and y in
  D(Vn) there exists an instantiation of the
  variables V1 ..Vn-1 such that all the constraint
  (Vi,Vi1) are satisfied
• CSP is path consistent iff each path is path
  consistent
• consistency along the path only

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Path Consistency (PC)
V2
V4
V3
V5
V0
V1
???

• checking paths of length 2 is enough
• Plus/Minus
• detects more inconsistencies than AC
• - extensional representation of constraints (01
  matrix), huge memory consumption
• - changes in graph connectivity

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K -consistency

• K-consistency
• consistent valuation o (K-1) variables can be
  extended to K-th variable
• strong K-consistency ? J-consistency for each
  J?K

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Is k-consistency enough ?

• If all the domains have cardinality 1gt solution
• If any domain is empty gt no solution
• Otherwise ?
• strongly k-consistent constraint graph with k
  nodes gt completeness
• for some special kind of graphs weaker forms of
  consistency are enough

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Consistency Completeness

• strongly N-consistent constraint graph with N
  nodes gt solution
• strongly K-consistent constraint graph with N
  nodes (KltN) gt ??? path consistent but no
  solution
• Special graph structures
• tree structured graph gt (D)AC is enough

?
A
D
1,2,3
1,2,3
?
?
?
?
C
B
?
1,2,3
1,2,3
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hyper-arc consistency

• A primitive constraint c is hyper-arc consistent
  with domain D if
• for each variable x in c and for each d in d(x)
  the valuation x--gtd can be extended to a solution
  of c
• A CSP is hyper-arc consistent if each prim.
  Constraint in it is hyper-arc consistent
• NC ? hyper-arc consistency for constraint with 1
  var
• AC ? hyper-arc consistency for constraint with 2
  var

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Non binary constraints

• What about prim. constraints with more than 2
  variables?
• Each CSP can be transformed into an equivalent
  binary CSP (dual encoding)
• k-consitncy
• hyper-arc consistency

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Dual encoding

• Each CSP can be transformed into an equivalent
  binary CSP by dual encoding
• k-ary constraint c is converted to a dual
  variable vc with the domain consisting of
  compatible tuples
• for each pair of constraint c,c sharing some
  variable there is a binary constraint between vc
  and vc restricting the dual variables to tuples
  in which the original shared variables take the
  same value

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Hyper-arc consistency

• hyper-arc consistency extending arc consistency
  to arbitrary number of variables
• (better than binary representation)
• Unfortunately determining hyper-arc consistency
  is NP-hard (as expensive as determinining if a
  CSP is satisfiable).
• Solution?

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Bounds Consistency

• arithmetic CSP domains integers
• range l..u represents the set of integers l,
  l1, ..., u
• idea use real number consistency and only examine
  the endpoints (upper and lower bounds) of the
  domain of each variable
• Define min(D,x) as minimum element in domain of
  x, similarly for max(D,x)

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Bounds Consistency

• A prim. constraint c is bounds consistent with
  domain D if for each var x in vars(c)
• exist real numbers d1, ..., dk for remaining vars
  x1, ..., xk such that min(D,xj), djlt max(D,xj)
  and
  is a solution of c
• and similarly for
• An arithmetic CSP is bounds consistent if all its
  primitive constraints are

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Bounds Consistency Examples
Not bounds consistent, consider Z2, then
X-3Y10 But the domain below is bounds consistent
Compare with the hyper-arc consistent domain
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Achieving Bounds Consistency

• Given a current domain D we wish to modify the
  endpoints of domains so the result is bounds
  consistent
• propagation rules do this

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Achieving Bounds Consistency
Consider the primitive constraint X Y Z
which is equivalent to the three forms
Reasoning about minimum and maximum values
Propagation rules for the constraint X Y Z
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Achieving Bounds Consistency
The propagation rules determine that
Hence the domains can be reduced to
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More propagation rules
Given initial domain
We determine that new domain
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Disequations
Disequations give weak propagation rules, only
when one side takes a fixed value that equals the
minimum or maximum of the other is there
propagation
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Multiplication
If all variables are positive its simple enough
Example becomes
But what if variables can be 0 or negative?
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Multiplication
Calculate X bounds by examining extreme values
Similarly for upper bound on X using maximum BUT
this does not work for Y and Z? As long as
min(D,Z) lt0 and max(D,Z)gt0 there is no bounds
restriction on Y
Recall we are using real numbers (e.g. 4/d)
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Multiplication
We can wait until the range of Z is non-negative
or non-positive and then use rules like
division by 0
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Bounds Consistency Algm

• Repeatedly apply the propagation rules for each
  primitive constraint until there is no change in
  the domain
• We do not need to examine a primitive constraint
  until the domains of the variables involve are
  modified

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Bounds Consistency Example
Smugglers knapsack problem (no whiskey available)
Continuing there is no further change Note how we
had to reexamine the profit constraint
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Bounds consistency solver

• D bounds_consistent(C,D)
• if D is a false domain
• return false
• if D is a valuation domain
• return satisfiable(C,D)
• return unknown

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Back. Bounds Cons. Solver

• Apply bounds consistency before starting the
  backtracking solver and after each variable is
  given a value

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Back. Bounds Solver Example
Smugglers knapsack problem (whiskey available)
Current domain
Initial bounds consistency
W 0
Solution Found return true
P 1
(0,1,3)
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Back. Bounds Solver Example
Smugglers knapsack problem (whiskey available)
Current domain
Initial bounds consistency
Backtrack
Backtrack
W 0
W 1
W 2
P 2
P 3
P 1
(1,1,1)
(2,0,0)
No more solutions
false
(0,1,3)
(0,3,0)
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Generalized Consistency

• Can use any consistency method with any other
  communicating through the domain,
• node consistency prim constraints with 1 var
• arc consistency prim constraints with 2 vars
• bounds consistency other prim. constraints
• Sometimes we can get more information by using
  complex constraints and special consistency
  methods

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Alldifferent

• alldifferent(V1,...,Vn) holds when each
  variable V1,..,Vn takes a different value
• alldifferent(X, Y, Z) is equivalent to
• Arc consistent with domain
• BUT there is no solution! specialized consistency
  for alldifferent can find it

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Alldifferent Consistency

• let c be of the form alldifferent(V)
• while exists v in V where D(v) d
• V V - v
• for each v in V
• D(v) D(v) - d
• DV union of all D(v) for v in V
• if DV lt V then return false domain
• return D

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Alldifferent Examples
DV 1,2, VX,Y,Z hence detect
unsatisfiability
DV 1,2,3,4,5, VX,Y,Z,T dont detect unsat.
Maximal matching based
consistency could
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Other Complex Constraints

• schedule n tasks with start times Si and
  durations Di needing resources Ri where L
  resources are available at each moment
• array access if I i, then X Vi and if X ! Vi
  then I ! i

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Optimization for CSPs

• Because domains are finite can use a solver to
  build a straightforward optimizer
• retry_int_opt(C, D, f, best)
• D2 int_solv(C,D)
• if D2 is a false domain then return best
• let sol be the solution corresponding to D2
• return retry_int_opt(C /\ f lt sol(f), D, f, sol)

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Retry Optimization Example
Smugglers knapsack problem (optimize profit)
First solution found
Next solution found
No next solution!
Corresponding solution
Return best solution
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Backtracking Optimization

• Since the solver may use backtrack search anyway
  combine it with the optimization
• At each step in backtracking search, if best is
  the best solution so far add the constraint f lt
  best(f)

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Back. Optimization Example
Smugglers knapsack problem (whiskey available)
Smugglers knapsack problem (whiskey available)
Current domain
Initial bounds consistency
W 0
Solution Found add constraint
P 1
(0,1,3)
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Back. Optimization Example
Smugglers knapsack problem (whiskey available)
Initial bounds consistency
W 1
W 0
W 2
P 2
P 3
(1,1,1)
P 1
false
(0,1,3)
false
false
Modify constraint
Return last sol (1,1,1)
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Branch and Bound Opt.

• The previous methods,unlike simplex don't use the
  objective function to direct search
• branch and bound optimization for (C,f)
• use simplex to find a real optimal,
• if solution is integer stop
• otherwise choose a var x with non-integer opt
  value d and examine the problems
• use the current best solution to constrain prob.

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Branch and Bound Example
Smugglers knapsack problem
false
false
Solution (2,0,0) 30
Solution (1,1,1) 32
false
Worse than best sol
false
false
false
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Finite Constraint Domains Summary

• CSPs form an important class of problems
• Solving of CSPs is essentially based on
  backtracking search
• Reduce the search using consistency methods
• node, arc, bound, generalized
• Optimization is based on repeated solving or
  using a real optimizer to guide the search