1. Foundations of Constraint Processing
Lookahead Schemas
Foundations of Constraint Processing
CSCE421/821, Spring 2008:
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 123B
Tel: +1(402)

2. Outline Looking ahead Schemas Forward checking (FC)
Directional Arc Consistency (DAC)
Maintaining Arc Consistency (a.k.a. full arc-consistency)

3. Looking ahead Rationale: Techniques
As decisions are made (conditioning),
Revise the domain of future variables to propagate the effects of decisions
i.e., eliminate inconsistent choices in future sub-problem
Domain annihilation of a future variable avoids expansion of useless portions of the tree
Techniques
Partial: forward-checking (FC), directional arc-consistency (DAC)
Full: Maintaining arc-consistency (MAC)
Use: Revise(Vf, Vc), Vf future variable, Vc current variable

4. Revising the domain of Vi
Revising the domain of Vi given a constraint CVi,Vj on Vi (i.e., Vi  Scope(C))
General notation: Revise(Vi,CVi,Vj)
In a binary CSP:
Revise(Vi,CVi,Vj)=Revise(Vi, Vj)

5. Revise(Vi, Vj) Revise(Vi, Vj) NOTE: only DVi may be updated
revised  nil
 x  DVi
 y  DVj
If  y  DVj with (x,y)  C return()
ElseIf revised  t DVi  DVi \ {x}
return(revised)
NOTE: only DVi may be updated

6. Domain filtering in lookahead
Vc current variable
Vf future variable
{Vf} all future variables
Revise(Vf, Vc)
FC(Vc):
 Vf  {Vf} connected to Vc
Revise(Vf,Vc)
If DVf ={} then return(failure)

7. Look-ahead techniques: FC, DAC, MAC
assumes a fixed variable ordering d
MAC:
does more pruning (search may visit fewer nodes) at the cost of more consistency checks
FC(Vc)
FC(Vc);
While not failure: For the next Vf in the ordering d, FC(Vf)
FC(Vc); AC({Vf})
FC(Vc);
Repeat until quiescence or failure
 Vf1,Vf2  {Vf}, Revise(Vf1,Vf2)

8. Terminology overload alert: FC
FC is used to denote any of the following:
a partial look-ahead schema
a specific chronological backtrack search algorithm that uses the partial look-ahead schema
Meaning is inferred from context
Not a healthy situation, but a fact of reality
Advice: state upfront the meaning of your terms and stick to them throughout your paper

9. (BT Search +) MAC vs. FC
Reference: [Sabin & Freuder, ECAI94], [Bessière & Régin, CP97], [Sabin & Freuder, CP97], [Gent & Prosser, APES ], [Experiments by Lin XU, 2001], [Yang, MS thesis 2003]
Results: (sketchy)
Low tightness
High tightness
Low density (sparse) FC
MAC
High density (dense)
Note: results depend on
Variable ordering (static vs. dynamic)
Problem difficulty (positive relative to crossover point)