Foundations of Constraint Processing

1
Intelligent Backtracking Algorithms

• Foundations of Constraint Processing
• CSCE421/821, Fall 2005
• www.cse.unl.edu/choueiry/F05-421-821/
• Berthe Y. Choueiry (Shu-we-ri)
• Avery Hall, Room 123B
• choueiry_at_cse.unl.edu
• Tel [phone]

2
Reading

• Required reading
• Hybrid Algorithms for the Constraint Satisfaction
  Problem Prosser, CI 93
• Recommended reading
• Chapters 5 and 6 of Dechters book
• Tsang, Chapter 5
• Notes available upon demand
• Notes of Fahiem Bacchus Chapter 2, Section 2.4
• Handout 4 and 5 of Pandu Nayak (Stanford)

3
Outline

• Review of terminology of search
• Hybrid backtracking algorithms
• Looking ahead
• Evaluation of (deterministic) BT search algorithms

4
Backtrack search (BT)

• Variable/value ordering
• Variable instantiation
• (Current) path
• Current variable
• Past variables
• Future variables
• Shallow/deep levels /nodes
• Search space / search tree
• Back-checking
• Backtracking

5
Outline

• Review of terminology of search
• Hybrid backtracking algorithms
• Vanilla BT
• Improving back steps BJ, CBJ
• Improving forward step BM, FC
• Looking ahead
• Evaluation of (deterministic) BT search algorithms

6
Two main mechanisms in BT

• Backtracking
• To recover from dead-ends
• To go back
• Consistency checking
• To expand consistent paths
• To move forward

7
Backtracking

• To recover from dead-ends
• Chronological (BT)
• Intelligent
• Backjumping (BJ)
• Conflict directed backjumping (CBJ)
• With learning algorithms (Dechter Chapt 6.4)
• Etc.

8
Consistency checking

• To expand consistent paths
• Back-checking against past variables
• Backmarking (BM)
• Look-ahead against future variables
• Forward checking (FC) (partial look-ahead)
• Directional Arc-Consistency (DAC) (partial
  look-ahead)
• Maintaining Arc-Consistency (MAC) (full
  look-ahead)

9
Hybrid algorithms

• Backtracking checking new hybrids

BT BJ CBJ
BM BMJ BM-CBJ
FC FC-BJ FC-CBJ

• Evaluation
• Empirical Prosser 93. 450 instances of Zebra
• Theoretical Kondrak Van Beek 95

10
Notations (in Prossers paper)

• Variables Vi, i in 1, n
• Domain Di vi1, vi2, ,viMi
• Constraint between Vi and Vj Ci,j
• Constraint graph G
• Arcs of G Arc(G)
• Instantiation order (static or dynamic)
• Language primitives list, push, pushnew, remove,
  set-difference, union, max-list

11
Main data structures

• v a (1xn) array to store assignments
• vi gives the value assigned to ith variable
• v0 pseudo variable (root of tree),
  backtracking to v0 indicates insolvability
• domaini a (1xn) array to store the original
  domains of variables
• current-domaini a (1xn) array to store the
  current domains of variables
• Upon backtracking, current-domaini of future
  variables must be refreshed
• check(i,j) a function that checks whether the
  values assigned to vi and vj are consistent

12
Generic search bcssp

• Procedure bcssp (n, status)
• Begin
• consistent ? true
• status ? unknown
• i ? 1
• While status unknown
• Do Begin
• If consistent
• Then i ? label (i, consistent)
• Else i ? unlabel (i, consistent)
• If i gt n
• Then status ? solution
• Else If i0 then status ? impossible
• End
• End

• Forward move x-label
• Backward move x-unlabel
• Parameters current variable, Boolean
• Return new current variable

13
Chronological backtracking (BT)

• Uses bt-label and bt-unlabel
• When vi is assigned a value from
  current-domaini, we perform back-checking
  against past variables (check(i,k))
• If back-checking succeeds, bt-label returns i1
• If back-checking fails, we remove the assigned
  value from current-domaini, assign the next
  value in current-domaini, etc.
• If no other value exists, vi-1 is
  un-instantiated and we seek a new value for it
  (notation in general vh)
• For all future variables j current-domainj
  domainj
• For all past variables g current-domaing ?
  domaing

14
BT-label

• Function bt-label(i,consistent) INTEGER
• BEGIN
• consistent ? false
• For vi ? each element of current-domaini
  while not consistent
• Do Begin
• consistent ? true
• For h ? 1 to (i-1) While consistent
• Do consistent ? check(i,h)
• If not consistent
• Then current-domaini ? remove(vi,
  current-domaini)
• End
• If consistent then return(i1) ELSE return(i)
• END

• Terminates
• consistenttrue, return i1
• consistentfalse, current-domaininil, returns i

15
BT-unlabel

• FUNCTION bt-unlabel(i,consistent)INTEGER
• BEGIN
• h ? i -1
• current-domaini ?domaini
• current-domainh ?remove(vh,current-domainh)
• consistent ? current-domainh ? nil
• return(h)
• END

• Is called when consistentfalse and
  current-domaininil
• Selects vh to backtrack to
• Uninstantiates all variables between vh and vi
• Removes vh from current-domain h
• Sets consistent to true if current-domainh ? 0
• Returns h

16
Example BT (the dumbest example ever)
v0
-
1,2,3,4,5
V1
v1
1
1,2,3,4,5
v2
V2
1
1,2,3,4,5
V3
v3
1
CV3,V4(V31,V43)
1,2,3,4,5
etc
V4
v4
2
1
3
4
CV2,V5(V25,V51),(V25,V54)
1,2,3,4,5
V5
v5
1
2
3
4
5
17
Outline

• Review of terminology of search
• Hybrid backtracking algorithms
• Vanilla BT
• Improving back steps BJ, CBJ
• Improving forward step BM, FC
• Looking ahead
• Evaluation of (deterministic) BT search algorithms

18
Danger of BT thrashing

• BT assumes that the instantiation of vi was
  prevented by a bad choice at (i-1).
• It tries to change the assignment of vi-1
• When this assumption is wrong, we suffer from
  thrashing (exploring barren parts of solution
  space)
• Backjumping (BT) tries to avoid that
• Jumps to the reason of failure
• Then proceeds as BT

19
Backjumping (BJ)

• Tries to reduce thrashing by saving some
  backtracking effort
• When vi is instantiated, BJ remembers vh, the
  deepest node of past variables that vi has
  checked against.
• Uses max-checki, global, initialized to 0
• At level i, when check(i,h) succeeds
• max-checki ?max(max-checki, h)
• If current-domainh is getting empty, simple
  chronological backtracking is performed from h
• BJ jumps then steps!

1
0
2
1
2
3
3
h-2
h-1
h-1
h
h
i
0
0
Past variable
0
Current variable
20
BJ label/unlabel

• bj-label same as bt-label, but updates
  max-checki
• bj-unlabel, same as bt-unlabel but
• Backtracks to h max-checki
• Resets max-checkj ? 0 for j in h1,i
• ? Important max-check is the deepest level we
  checked against, could have been success or could
  have been failure

21
Example BJ
v0 0
-
v1
Max-check1 0
1
v2
2
1
Max-check2 1
v3
1
V41, fails for V2, mc2
V42, fails for V2, mc2
V43, succeeds
v4
2
1
3
4
max-check4 3
V51, fails for V1, mc1
V52, fails for V2, mc2
V53, fails for V1
v5
V54, fails for V1
1
2
3
4
5
V55, fails for V1
max-check5 2
22
Conflict-directed backjumping (CBJ)

• Backjumping
• jumps from vi to vh,
• but then, it steps back from vh to vh-1 ?
• CBJ improves on BJ
• Jumps from vi to vh
• And jumps back again, across conflicts involving
  both vi and vh
• To maintain completeness, we jump back to the
  level of deepest conflict

23
CBJ data structure
conf-set
0
1
2

• Maintains a conflict set conf-set
• conf-seti are first initialized to 0
• At any point, conf-seti is a subset of past
  variables that are in conflict with i

g
0
conf-setg
h-1
0
conf-seth
h
0
conf-seti
i
0 0 0
24
CBJ conflict-set

• When a check(i,h) fails
• conf-seti ? conf-seti ?h
• When current-domaini empty
• Jumps to deepest past variable
• h in conf-seti
• Updates
• conf-seth ? conf-seth ? (conf-seti \h)
• Primitive form of learning (while searching)

25
Example CBJ
26
Backtracking summary

• Chronological backtracking
• Steps back to previous level
• No extra data structures required
• Backjumping
• Jumps to deepest checked-against variable, then
  steps back
• Uses array of integers max-checki
• Conflict-directed backjumping
• Jumps across deepest conflicting variables
• Uses array of sets conf-seti

27
Outline

• Review of terminology of search
• Hybrid backtracking algorithms
• Vanilla BT
• Improving back steps BJ, CBJ
• Improving forward step BM, FC
• Looking ahead
• Evaluation of (deterministic) BT search algorithms

28
Backmarking goal

• Tries to reduce amount of consistency checking
• Situation
• vi about to be re-assigned k
• vi ? k was checked against vh?g
• vh has not been modified

vh g
vi
k
29
BM motivation

• Two situations
• Either (vik,vhg) has failed ? it will fail
  again
• Or, (vik,vhg) was founded consistent ? it
  will remain consistent

• In either case, back-checking effort against vh
  can be saved!

30
Data structures for BM 2 arrays

• maximum checking level mcl (n x m)
• Minimum backup level mbl (n x 1)

31
Maximum checking level

• mcli,k stores the deepest variable that vi?k
  checked against
• mcli,k is a finer version of max-checki

32
Minimum backup level

• mbli gives the shallowest past variable whose
  value has changed since vi was the current
  variable

• BM (and all its hybrid) do not allow dynamic
  variable ordering

33
When mcli,kmblij

• BM is aware that
• The deepest variable that (vi ?k) checked
  against is vj
• Values of variables in the past of vj (hltj)
  have not changed
• So
• We do need to check (vi ?k) against the values
  of the variables between vj and vi
• We do not need to check (vi ?k) against the
  values of the variables in the past of vj

vj
vi
k
mbli j
34
Type a savings
When mcli,k lt mbli, do not check vi ? k
because it will fail
35
Type b savings

• When mcli,k ? mbli, do not check (i,hltj)
  because they will succeed

h
vj
vg
vi
k
mcli,k?mbli
mcli,kg mbli j
36
Hybrids of BM

• mcl can be used to allow backjumping in BJ
• Mixing BJ BM yields BMJ
• avoids redundant consistency checking (types ab
  savings) and
• reduces the number of nodes visited during search
  (by jumping)
• Mixing BM CBJ yields BM-CBJ

37
Problem of BM and its hybrids warning
BMJ enjoys only some of the advantages of
BM Assume mblh m and max-checkimax(mcli
,x)g

• Backjumping from vi
• vi backjumps up to vg
• Backmarking of vh
• When reconsidering vh, vh will be checked
  against all f in m,g(
• effort could be saved ?
• Phenomenon will worsen with CBJ
• Problem fixed by Kondrak van Beek 95

vm
vm
vf
vg
vh
vi
38
Forward checking (FC)

• Looking ahead from current variable, consider
  all future variables and clear from their domains
  the values that are not consistent with current
  partial solution
• FC makes more work at every instantiation, but
  will expand fewer nodes
• When FC moves forward, the values in
  current-domain of future variables are all
  compatible with past assignment, thus saving
  backchecking
• FC may wipe out the domain of a future variable
  (aka, domain annihilation) and thus discover
  conflicts early on. FC then backtracks
  chronologically
• Goal of FC is to fail early (avoid expanding
  fruitless subtrees)

39
FC data structures

• When vi is instantiated, current-domainj are
  filtered for all j connected to i and iltjltn
• reductionj store sets of values remove from
  current-domainj by some variable before vj
• reductionsj a, b, c, d, e, f, g, h
• future-fci subset of the future variables that
  vi checks against (redundant)
• future-fci k, j, n
• past-fci past variables that checked against
  vj
• All these sets are treated like stacks

40
Forward Checking functions

• check-forward
• undo-reductions
• update-current-domain
• fc-label
• fc-unlabel

41
FC functions

• check-forward(i,j) is called when instantiating
  vi
• It performs REVISE(j,i)
• Returns false if current-domainj is empty, true
  otherwise
• Values removed from current-domainj are pushed,
  as a set, into reductionsj
• These values will be popped back if we have to
  backtrack over vi (undo-reductions)

42
FC functions

• update-current-domain
• current-domaini ? domaini \ reductionsi
• actually, we have to iterate over reductionsset
  of sets
• fc-label
• Attempts to instantiate current-variable
• Then filters domains of all future variables
  (push into reductions)
• Whenever current-domain of a future variable is
  wiped-out
• vi is un-instantiated and
• domain filtering is undone (pop reductions)

43
Hybrids of FC

• FC suffers from thrashing it is based on BT
• FC-BJ
• max-check is integrated in fc-bj-label and
  fc-bj-unlabel
• Enjoys advantages of FC and BJ but suffers
  malady of BJ (jump the step)
• FC-CBJ
• Best algorithm for far (assuming static variable
  ordering)
• fc-cbj-label and fc-cbj-unlabel

44
Consistency checking summary

• Chronological backtracking
• Uses back-checking
• No extra data structures
• Backmarking
• Uses mcl and mbl
• Two types of consistency-checking savings
• Forward-checking
• Works more at every instantiation, but expands
  fewer subtrees
• Uses reductionsi, future-fci, past-fci

45
Experiments

• Empirical evaluations on Zebra
• Representative of design/scheduling problems
• 25 variables, 122 binary constraints
• Permutation of variable ordering yields new
  search spaces
• Variable ordering different bandwidth/induced
  width of graph
• 450 problem instances were generated
• Each algorithm was applied to each instance
• Experiments were carried out under static
  variable ordering

46
Analysis of experiments

• Algorithms compared with respect to
• Number of consistency checks (average)
• FC-CBJ lt FC-BJ lt BM-CBJ lt FC lt CBJ lt BMJ lt BM lt
  BJ lt BT
• Number of nodes visited (average)
• FC-CBJ lt FC-BJ lt FC lt BM-CBJ lt BMJ BJ lt BM BT
• CPU time (average)
• FC-CBJ lt FC-BJ lt FC lt BM-CBJ lt CBJ lt BMJ lt BJ lt
  BT lt BM
• FC-CBJ apparently the champion

47
Additional developments

• Other backtracking algorithms exist
• Graph-based backjumping (GBJ), etc. Dechter
• Pseudo-trees Freuder 85
• Other look-ahead techniques exist
• DAC, MAC, etc.
• More empirical evaluations
• over randomly generated problems
• Theoretical evaluations
• Based on approach of Kondrak Van Beek IJCAI95

48
Outline

• Review of terminology of search
• Hybrid backtracking algorithms
• Looking ahead
• Forward checking (FC)
• Directional Arc Consistency (DAC)
• Maintaining Arc Consistency (a.k.a. full
  arc-consistency)
• Evaluation of (deterministic) BT search algorithms

49
Looking ahead

• Rationale
• 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

50
Revising the domain of Vi

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

51
Revise(Vi, Vj)

• Revise(Vi, Vj)
• ? x ? DVi
• ? y ? DVj
• If ? y ? DVj with (x,y) ? C return()
• ElseIf DVi ? DVi \ x
• NOTE only DVi may be updated

52
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)

53
Look-ahead techniques FC, DAC, MAC

• FC
• DAC
• assumes a fixed variable ordering d
• MAC
• does more pruning at the cost of more consistency
  checks

FC(Vc)
FC(Vc) ? Vf ? Vf, FC(Vf) in the ordering d
FC(Vc) AC(Vf)
FC(Vc) Repeat until quiescence or failure
? Vf1,Vf2 ? Vf, Revise(Vf1,Vf2)
54
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 reality
• Advice state upfront the meaning of your terms
  and stick to them throughout your paper

55
Outline

• Review of terminology of search
• Hybrid backtracking algorithms
• Looking ahead
• Evaluation of (deterministic) BT search
  algorithms Dechter, 6.6.2
• CSP parameters
• Comparison criteria
• Theoretical evaluations
• Empirical evaluations

56
CSP parameters

• Number of variables n
• Domain size a, d
• Constraint tightness
• t forbidden tuples / all tuples
• Proportion of constraints (a.k.a., constraint
  density, constraint probability)
• p1 e / emax, e is nbr of constraints

57
Comparison criteria

• Number of nodes visited (NV)
• Every time you call label
• Number of constraint check (CC)
• Every time you call check(i,j)
• CPU time
• Be as honest and consistent as possible
• Some specific criterion for assessing the quality
  of the improvement proposed
• Presentation of values
• Average or median of criterion
• (qualified) run-time distribution
• Solution-quality distribution

58
Theoretical evaluations

• Comparing NV and/or CC
• Common assumptions
• for finding all solutions
• static orderings

59
Empirical evaluation data sets

• Use real-data sets (anecdotal evidence)
• Use benchmarks (csp library)
• Use randomly generated problems

60
Empirical evaluations random problems

• Various models exist (use Model B)
• Models A, B, C, E, F, etc.
• Vary parameters ltn, a, t, pgt
• Number of variables n
• Domain size a, d
• Constraint tightness t forbidden tuples /
  all tuples
• Proportion of constraints (a.k.a., constraint
  density, constraint probability) p1 e / emax
• Issues
• Uniformity
• Difficulty (phase transition)
• Solvability of instances (for incomplete search
  techniques)

61
Model B

1. Input n, a, t, p1
2. Generate n nodes
3. Generate a list of n2 tuples of all combinations
   of 2 nodes
4. Choose e elements from above list as constraints
   to between the n nodes
5. If the graph is not connected, throw away, go
   back to step 4, else proceed
6. Generate a list of a2 tuples of all combinations
   of 2 values
7. For each constraint, choose randomly a number of
   tuples from the list to guarantee tightness t for
   the constraint

62
Example MAC vs. FC

• Reference Sabin Freuder, ECAI94, Bessière
  Régin, CP97, Sabin Freuder, CP97, Gent
  Prosser, APES-20-2000, Experiments by Lin XU,
  2001, Yang, MS thesis 2003
• Results (sketchy)

Low tightness High tightness
Low density (sparse) FC MAC
High density (dense) FC FC

• Note results depend on
• Variable ordering (static vs. dynamic)
• Problem difficulty (positive relative to
  crossover point)