s 1

Download Report

Transcript s 1 

Structural Decomposition
Methods of CSPs
• Input: a constraint hypergraph
• Output: an equivalent join tree
s2
s3
1 4
s1
0
s5
5
s6
6
s7
7
s8 s9
8 9 10 19
s10
s17
18 20 s16
22
3 11 12 13 14 15 16 17 21 s
15
s4 s11 s12 s13 s14
2
A constraint hypergraph Hcg
associated with a CSP
s1
s2
s3
s4
s5
s6
s11
s12
s10
s7
s13
s8
s9
s14
s16
s15
s17
An equivalent join tree of Hcg
Criteria to compare structural decomposition methods:
(1) CPU time for decomposition (2) Width of the join tree
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
1
Contribution in Context
HYPERTREE[1]
TRAVERSE
CaT
CUT
HINGE+
HINGETCLUSTER[5]
HYPERCUTSET[11]
HINGE[5] TCLUSTER[8]  w*[6]
= TREEWIDTH [9]
BICOMP[3]
CUTSET[4]
HINGE+: An improvement to HINGE.
CUT: A variation of HINGE+.
TRAVERSE: Based on a simple sweep of the constraint hypergraph.
CaT: A combination of CUT and TRAVERSE.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
2
i-cut
s2
s3
1 4
s1
0
s5
5
s6
6
s7
7
s8 s9
8 9 10 19
s10
s17
18 20 s16
22
3 11 12 13 14 15 16 17 21 s
15
s4 s11 s12 s13 s14
2
A 2-cut: {S6, S12}.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
3
Hinge decomposition (HINGE)
HINGE
Input:
A constraint hypergraph H.
Output: An equivalent join tree T of H.
The edges of T are labeled.
Process: Continuously finds 1-cuts in H.
s11 s11 s17
s1 s
2
s2
Applying
HINGE to
Hcg
s2 s3
s4 s5 s6 s9 s s
9 10
s7 s8 s9
s9
s11 s12
s9 s15
s13 s14
s9
s9 s16
Width = 12
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
4
Hinge+ Decomposition (HINGE+)
HINGE+
Input:
a constraint hypergraph H and a maximum cut size k.
Output: An equivalent join tree T of H. The edges of T are labeled.
Process: Finds 1-cuts through k-cuts in H.
When there are multiple possible i-cuts, chooses the one that yields the
best division (i.e., the size of the largest sub-problem is the smallest).
s1 s s s s
2
2 3 4
s2
s4 s5 s5
Applying
HINGE+
to Hcg,
k=2
s4
s5 s6
s11 s12
s11
s6 s6 s7 s7
s12 s12 s13 s13
s9 s9 s10
s7
s
s8 s8 s8 s9
s9 s15
s13 s14 s 9
14
s14
s9 s s
9 16
s11 s17
Width = 5
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
5
Cut Decomposition (CUT)
CUT: A variation of HINGE+.
Input:
A constraint hypergraph H and a maximum cut size k.
Output: An equivalent join tree T of H. The edges of T are labeled.
Process: Finds 1-cuts through k-cuts in H.
CUT guarantees every tree node of T contains at most 2 different cuts.
When there are multiple possible i-cuts that satisfies the condition,
choose the one that yields the best division.
Applying
CUT to
Hcg,
k=2
s1 s s s s
2 3 3
2
s2
s4 s4
s3
s4 s5 s5 s6 s6
s5 s11 s11 s12 s12
s11
s6 s12
s6 s12 s17
s6
s7 s7
s12 s13
s13
s9 s9 s10
s6
s
s7 s8 s8 s9 s s
9 15
s12 s14 s 9
14
s13
s9 s s
9 16
Width = 4
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
6
Traverse Decomposition:
TRAVERSE-I
TRAVERSE-I
Input:
A constraint hypergraph H = (V, E) and a set F  E.
Output: An equivalent join tree T of H.
Process: Sweep through the constraint hypergraph from F.
Applying
TRAVERSE-I
to Hcg,
F = {s1}
s1
s2
s3
s4
s5
s11
s6
s12
s17
s7
s13
s8
s14
s9
s10
s15
s16
Width = 3
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
7
TRAVERSE-II
TRAVERSE-II
Input:
A constraint hypergraph H = (V, E) and two sets F1, F2  E.
Output: An equivalent join tree T of H.
Process: Sweep through the constraint hypergraph from F1 to F2.
Applying
TRAVERSE-II
to Hcg,
F1 = {s1, s2}
F2 = {s9, s16}
s1
s2
s3
s4
s5
s11
s6
s12
s17
s7
s13
s8
s14
s9
s10
s15
s9
s16
Width = 3
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
8
Cut-and-Traverse Decomposition
(CaT)
CaT: A combination of CUT and TRAVERSE.
Input:
A constraint hypergraph H and a maximum cut size k.
Output: An equivalent join tree T of H.
Process: (1) Apply CUT to H and get a join tree Tm
(2) For every tree node Ni inTm ,
a. If it contains no cut, apply TRAVERSE-I to Ni from an
arbitrary hyperedge.
b. If it contains one cut C1, apply TRAVERSE-I to Ni fromC1.
c. If it contains two cuts C1 and C2, apply TRAVERSE-II from C1 to C2.
Applying CaT
to Hcg,
K = 2.
s1
s2
s5
s11
s3
s4
s6
s12
s17
s10
s7
s13
s8
s14
s9
s16
s15
Width = 2
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
9
Preliminary Experiments
# Constraints = 20, Maximum arity = 4.
For HINGE+, CUT, and CaT, maximum cut size is 2.
Average CPU time
(millseconds)
Comparing CPU time
30.00
25.00
20.00
15.00
10.00
5.00
0.00
HINGE
HINGE+
CUT
CaT
TRAVERSE
27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Variable Number
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
10
Preliminary Experiments
Average Width
Comparing width
13.00
11.00
9.00
7.00
5.00
3.00
1.00
-1.00
HINGE
HINGE+
CUT
CaT
TRAVERSE
27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Variable Number
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
11
Conclusions
1.
2.
3.
4.
5.
All these decomposition methods can be performed in polynomial
time.
• HINGE+ O(|V||E|k+1)
• CUT O(|V||E|k+1)
• TRAVERSE O(|V||E|2)
• CaT O(|V||E|k+1)
• k is the maximum cut size
HINGE+ strongly generalizes HINGE.
CaT strongly generalizes CUT.
HYPERTREE strongly generalizes HINGE+, CUT, TRAVERSE,
and CaT.
CaT experimentally decomposes better than HINGE, HINGE+,
CUT, and TRAVERSE.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
12
Future Work
•
•
More thorough experiments on randomly
generated constraint hypergraphs.
Compare CaT with HINGETCLUSTER and
HINGEHYPERTREE.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
13
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Gottlob, G., Leone, N., Scarcello, F. : On Tractable Queries and Constraints. In: 10th International Conference
and Workshop on Database and Expert System Applications (DEXA 1999). (1999)
Decther, R.: Constraint Processing. Morgan Kaufmann (2003)
Freuder, E.C.: A Sufficient Condition for Backtrack-Bounded Search. JACM 32 (4) (1985)
Dechter, R.: Constraint networks, Encyclopedia of Artificial Intelligence, 2nd edition, Wiley. New York, PP.276285. (1992)
Gyssens, M., Jeavons, P.G., Cohen, D.A.: Decomposing Constraint Satisfaction Problems using Database
Techniques. Artificial Intelligence 38 (1989)
Jeavons, P.G., Cohen, D.A., Gyssens, M. : A structural Decomposition for Hypergraphs. Contemporary
Mathematics 178 (1994)
Dechter, R., Pearl. J: Network based heuristic for constraint satisfaction problems, Artificial Intelligence 34 (1)
pp 1-38. (1988)
Decther, R., Pearl. J: Tree Clustering for Constraint Networks. Artificial Intelligence 38 (1998) Gottlob, G.,
Leone, N., Scarcello, F.: Hypertree Decompositions and Tractable Queries. Journal of Computer and System
Sciences 64. (2002)
Robertson, N., Seymour, P.D., Graph Minors II. Algorithmic aspects of tree width, J. Algorithms 7 309-322.
(1986)
Harvey, P., Ghose, A.: Reducing Redundancy in the Hypertree Decomposition Scheme. IEEE International
Conference on Tools with Artificial Intelligence (ICTAI 03). (2003)
Gottlob, G., Leone, N., Scarcello, F.: A comparison of Structural CSP Decomposition Methods. Artificial
Intelligence 124 (2000)
Gottlob, G., Hutle, M., Wotawa, F.: Combining Hypertree, Bicomp, And Hinge Decomposition. ECAI 02 (2002)
Zheng, Y., Choueiry B.Y.: Cut-and-Traverse: A New Structural Decomposition Strategy for Finite Constraint
Satisfaction Problems. CSCLP 04 (2004).
This research is supported by CAREER Award #0133568 from
the National Science Foundation.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
14