towards theoretical frameworks for comparing constraint satisfaction models and algorithms peter van...
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Towards theoretical frameworks for comparing constraint satisfaction models and algorithms
Peter van Beek, University of WaterlooCP 2001 · Paphos, CyprusNovember 2001
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Acknowledgements
Joint work with:
Fahiem Bacchus
Xinguang Chen
Grzegorz Kondrak
Toby Walsh
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Constraint programming methodology
Model problem·specify in terms of constraints on acceptable solutions
·define/choose constraint model: variables, domains, constraints
Solve model·define/choose algorithm·define/choose heuristics
Verify and analyze solution
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Previous work
Many models, algorithms, and heuristics proposed
Choice of model, algorithm, and heuristic can greatly influence efficiency
(e.g., Nadel, 1990; Ginsberg et al., 1990;
Frost and Dechter, 1994; Tsang et al., 1995 Smith et al., 2000; Beacham et al., 2001)
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How to compare?
Empirical studies·benchmark sets·random problems
Theoretical studies·worst case·average case
(e.g. Brown & Purdom, 1981;
Nadel, 1983; Wilf, 1984)
·pair-wise comparisons-partial orders-bounded worse
Performance measures
·nodes visited·constraint checks·CPU time
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Part I
Comparing backtracking algorithms
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Some backtracking algorithms
Chronological Backtracking (BT)
Backjumping (BJ)(Gaschnig, 1978)
Conflict-Directed Backjumping (CBJ)(Prosser, 1993)
Forward Checking (FC)(Haralick & Elliott, 1980; McGregor, 1979)
Maintaining arc consistency (MAC)(Haralick & Elliott, 1980; McGregor, 1979; Sabin & Freuder, 1994)
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Example: 4-queens
4
3
2
1
x1 x2 x3 x4
xi xj and
| xi - xj | | i - j |
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Search tree for 4-queens
x1
x2x3
x4
1 2 3 4
(1,1,1,1) (4,4,4,4)(2,4,1,3) (3,1,4,2)
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4
3
2
1
{x1 1}
Backjumping
Q
Q{x1 1, x2
3}
x1
x2
x1
x2
{x1 1, x2 3 , x4
2}
Q
x3
x1 x3 x4x2
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“New” algorithms: BJ1, …, BJn
BJk :·allowed to backjump at most k times from a leaf·after that it must chronologically backtrack
Special cases:·BJ1 equivalent to BJ·BJn equivalent to CBJ
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k-consistency
Given ·any k-1 distinct variables,·any consistent assignment to those variables,·any additional variable x
there exists a consistent assignment to x
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3
2
1
x1 x2 x3 x4
2-consistent ?
Q
Q1-consistent ?
3-consistent ?strongly 2-
consistent
yes
yes
no
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Maintaining consistency
{ x1 1, x2
4 }
4
3
2
1
x1 x2 x3 x4
Q
Qx3 x4
|x3 - x4| 1
Binary constraints
Unary constraintsx3 1, 3, 4
x4 1, 2, 4
Induced CSP
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Maintaining consistency
4
3
2
1
x1 x2 x3 x4
Q
Q
{ x1 1, x2
4 }
x3 x4
|x3 - x4| 1
Binary constraints
Unary constraintsx3 1, 3, 4
x4 1, 2, 4
FC: strong 1-consistency
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Maintaining consistency
Q
Q
{ x1 1, x2
4 }
x3 x4
|x3 - x4| 1
Binary constraints
Unary constraintsx3 1, 3, 4
x4 1, 2, 4
4
3
2
1
x1 x2 x3 x4
MAC: strong 2-consistency
Q
Q
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“New” algorithms: MC1, …, MCn
MCk :·maintains strong k-consistency on induced CSP at each node in backtrack tree
Special cases:·MC1 equivalent to FC·MC2 equivalent to MAC for binary CSPs
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Comparing backtracking algorithms
Proposal:for all CSP models for all variable orderings
algorithm a1 algorithm a2
Desirable features:·still holds if use best possible CSP model for a2
·still holds if use best possible variable ordering for a2
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Theoretical methodology
Using local consistency concepts:·formulate necessary and sufficient conditions for a node to be visited by a backtracking algorithm
Using conditions:·construct partial orders of the algorithms
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4
3
2
1
x1 x2 x3 x4
{x1 1}
{x1 1, x2
4}?
Example sufficient condition: FC
If a node is consistent and its parent is strongly 1-consistent, then FC visits the node
Q
Q
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Example necessary condition: MAC-CBJ
If MAC-CBJ visits a node, then it is consistent and its parent is strongly 2-consistent
4
3
2
1
x1 x2 x3 x4
{x1 1}
{x1 1, x2
4}?
Q
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Example conclusion: MAC-CBJ vs FC
FC visits p
MAC-CBJ visits p
p is consistent and parent(p) strongly 1-consistent
p is consistent and parent(p) strongly 2-consistent
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Partial order for nodes visited
MCk+1-CBJ
MCk-CBJ
MC1-CBJ
MCn-CBJ
MCk+1
MCk
MC1
MCn
BJk+1
BJk
BJ1
BJn
BT
...
...
...
...
...
...
less than or equal
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Partial order for nodes visited
MCk+1-CBJ
MCk-CBJ
MC1-CBJ
MCn-CBJ
MCk+1
MCk
MC1
MCn
BJk+1
BJk
BJ1
BJn
BT
...
...
...
......
...
incomparable
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Partial order for nodes visited
BT
MCk+1-CBJ
MCk-CBJ
MC1-CBJ
MCn-CBJ
MCk+1
MCk
MC1
MCn
BJk+1
BJk
BJ1
BJn
...
...
...
......
...
incomparable
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Partial order for nodes visited
MCk+1-CBJ
MCk-CBJ
MC1-CBJ
MCn-CBJ
MCk+1
MCk
MC1
MCn
BJk+1
BJk
BJ1
BJn
BT
...
...
...
...
...
...
incomparable
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Partial order for nodes visited
MCk+1-CBJ
MCk-CBJ
MC1-CBJ
MCn-CBJ
MCk+1
MCk
MC1
MCn
CBJBT
...
......
...
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Closer look: nodes visited, binary CSPs
MAC-CBJ
FC-CBJ
MAC
FC
CBJ
BJ2
BJ
BT
less than or equal
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Closer look: constraint checks, binary CSPs
MAC-CBJ
FC-CBJ
MAC
FC
CBJ
BJ2
BJ
BT
less than or equal
at most O(f) more
O(nd)
O(n2d2)
O(nd)
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Experiments: lookahead
MAC vs FC Faster? CommentsMcGregor (1979) FC 3 fasterHaralick & Elliott (1980) FC 3 fasterSabin & Freuder (1994) MAC much betterBacchus & van Run (1995) FC 3 20 fasterBessiere & Regin (1996) MAC much betterLarrosa (2000) both much better
Path vs MAC Faster? CommentsBacchus & van Run (1995) MAC much better
FC
MAC
O(nd)
Path
BT
O(nd)
O(nd)
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Example: FC vs MAC
Adapted from J. Ullman (1988); suggested by D. De Haan
Odd-even relation
x y1 21 42 12 33 23 44 14 3
variables: x1, …, xn, n odd
domains: {1, 2, 3, 4}
x1 x2 xn...
FC: 2n+1 - 4
MAC: 4
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Increasing lookahead
Problem MC2 MC3 MC4
1 > 3600 262 602 794 40 1403 7 3 34 1065 21 25 34 17 26 > 3600 125 37 58 6 48 17 3 29 163 95 210 87 4 2
Time (sec.) to solve 6 x 6 puzzlesMC4
MC3
MC2
MC1
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Experiments: lookback
FC vs FC-CBJ Faster? CommentsProsser (1993) FC-CBJ three times betterFrost & Dechter (1994) FC-CBJ somewhat betterBacchus & van Run (1995) FC-CBJ slightlySmith & Grant (1995) FC-CBJ sometimes much betterBayardo & Schrag (1996, 1997) FC-CBJ much betterLi & Anbulagan (1997) FC as good or better
FC-CBJFC
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Experiments: lookahead vs lookback
Increasing the level of consistency decreases effectiveness of CBJ
·Prosser (1993)·Bacchus & van Run (1995) ·Bessiere & Regin (1996)
MAC much better than FC-
CBJ
MAC better than MAC-CBJ
MAC-CBJMAC
FC-CBJFC
O(nd)
CBJBT
O(nd)
“CBJ becomes useless”
· Jussien, Debruyne, Boizumault (2000)MAC-DBT much better than MAC (binary)
·Chen & van Beek (2001)MGAC-CBJ much better than MAC (non-binary)
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Theory: lookahead vs lookback
MCk+1-CBJ
MCk-CBJ
MCk+1
MCk
BJk+1
BJk
BJk+1 better only if there exists a (k+1)-level backjump that is not a chronological backtrack
Number of (k+1)-level backjumps number of k-level backjumps
Therefore, as k increases, effectiveness of backjumping decreases
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Part II
Comparing CSP models
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Conversion to binary
Any non-binary CSP can be converted into one with only binary constraints
·dual graph method (Dechter & Pearl, 1989; Rossi et al. 1989)
·hidden variable method (Peirce, 1933; Rossi et al. 1989; Dechter 1990)
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Crossword puzzles
1 2 3
6
4
7
5
8
10
9
20
11
22
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21
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17
14
181615
23
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aaardvarkabackabacusabaftabaloneabandon...
Mona Lisamonarchmonarchymonarda...zymurgyzyrianzythum
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Non-binary model (original)
1 2 3
6
4
7
5
8
10
9
20
11
22
12
21
13
17
14
181615
23
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variables: one for each
unknown letter (cell)
domains: ‘a’, …, ‘z’
constraints: contiguous
letters must form words in dictionary
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Dual model (binary)
1 2 3
6
4
7
5
8
10
9
20
11
22
12
21
13
17
14
181615
23
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variables: one for each unknown word across and down
domains: words from
dictionary
constraints intersecting words must agree on common letter
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Hidden model (binary)
1 2 3
6
4
7
5
8
10
9
20
11
22
12
21
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17
14
181615
23
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variables: one for each unknown letter (cell) and
one for each unknown word across and down
domains: letters, words
constraints: letter and word
variable must agree
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Comparing models
One proposal:for all backtracking algorithms for all variable orderings
model m1 model m2
Objections:·effectiveness of a model is algorithm dependent
·models may contain different variables
Desirable features:·meets objections·still holds if use best possible variable ordering for m2
Another proposal:given backtrack algorithm for all v2 for m2
there exists a v1 for m1 s. t. model m1 model m2
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Theoretical methodology
Bound difference in number of nodes visited·compare pruning power of local consistencies of alternative models
(e.g., Debruyne & Bessiere, 1997;
Stergiou & Walsh, 1999; Prosser et al.,
2000; Walsh, 2000)·establish correspondence between nodes
Bound difference in cost at each node:·local consistency·variable ordering
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Some relationships
MGAC-dual MGAC-hiddenMGAC-orig
FC-dual FC-hiddenFC-orig
BT-dual BT-hiddenBT-orig
polynomial not polynomial
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Part III
Comparing variable ordering heuristics
Future work
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Conclusion
Technical:·comparing backtracking algorithms
partial order·comparing CSP models
bounded worse relation on algorithm-model combinations
Big picture:·theory can guide, inform experimentation