1. Improving the Performance of Consistency Algorithms by Localizing and Bolstering Propagation in a Tree Decomposition
Shant Karakashian, Robert J. Woodward and Berthe Y. Choueiry
Constraint Systems Laboratory, University of Nebraska-Lincoln, USA
Contributions
Bolstering Propagation at Separators
Empirical Evaluation
Localizing consistency to the clusters of a tree decomposition of a CSP & bolstering propagation at the separators between clusters
Theoretical characterization of resulting new consistency properties
Empirical evaluation of our approach establishing its benefits on difficult benchmarks, solving many problems in a backtrack-free manner and, thus, approaching ‘practical tractability’
Localizing R(∗,m)C
+ maxRPWC, m=3,4
wR(∗,2)C
R(∗,|ψ(cli)|)C
#inst.
GAC
global
local
Proj.
binary
clique
Completed
UNSAT
479
167
34.9%
170
35.5%
172
35.9%
169
35.3%
162
33.8%
285
59.5%
286
59.7%
282
58.9%
271
56.6%
SAT
200
174
87.0%
179
89.5%
178
89.0%
176
88.0%
84.5%
104
52.0%
152
76.0%
138
69.0%
124
62.0%
113
56.5%
BT-Free
0.0% 70
14.6% 39
8.1% 74
15.4%
187
39.0%
223
46.6%
213
44.5% 44
22.0% 55
27.5% 37
18.5% 53
26.5% 52
26.0% 38
19.0%
19.5% 77
38.5% 71 58
29.0%
Min(#NV) 17
3.5% 73
15.2% 43
9.0% 72
15.0%
16.1%
220
45.9%
249
248
51.8%
239
49.9% 47
23.5% 64
32.0% 62
31.0% 61
30.5% 83
41.5%
111
55.5%
100
50.0% 79
39.5%
Fastest 13
2.7% 35
7.3% 5
1.0% 1
0.2%
36.7%
108
22.5% 42
8.8%
7.7%
121
60.5% 45 23
11.5% 14
7.0% 12
6.0% 34
17.0% 18
6.5%
We localize the application of the consistency algorithm to each cluster of the tree decomposition, which allows us to increase the value of m to the number of relations in the cluster and, thus, the level of consistency enforced.
Bolstering Separators
We bolster constraint propagation along the tree by adding redundant constraints to the separators.
A perfect ‘communication’ between clusters requires a unique constraint over the separator’s variables, but materializing such a constraint is prohibitive in terms of space [Fattah and Dechter 1996; Kask+ 2005].
We propose three approximation schemes to this end: constraint projection (proj), binary constraints (bin), clique constraints (clq).
Graphical Representation
Hypergraph
Dual graph
Triangulated primal graph & its maximal cliques A B C E D F G H I J K M L N R4 R2 R3 R6 R7 R1 R7 R2 R3 R5 R6 R1
A,B,C,N
I,J,K
I,M,N
A,K,L
F,G,H
B,D,E,F
C,D,H N A I K H F B C R4 D A B C E D F G H I J K M L N C1 C2 C7 C3 C4 C5 C6 C8 C9
C10 R5
Two adjacent clusters
Unique constraint
Constraint projections
Tree decomposition E R6 R5 R7 R4 R2 R1 R3 B A D C F E R6 R5 R7 R4 R2 R1 R3
R3’ B A D C F
{A,B,C,N},{R1}
{A,I,N},{}
{B,C,D,H},{R6}
{I,M,N},{R2}
{B,D,F,H},{} C1 C2 C3 C7 C8
{A,I,K},{} C4
{I,J,K},{R3} C5
{A,K,L},{R4} C6
{B,D,E,F},{R5} C9
{F,G,H},{R7}
C10 E F R6 R5 R7 R2 R1 R3 B A D C
Rsep
Relational Consistency
R(∗,m) ensures that subproblem induced in the dual CSP by every connected combination of m relations is minimal [Karakshian+ 2010] R4 R7 R3 R6 R2 R1 R5
Primal graph separator…
Binary constraints
Clique constraints
Cumulative count of instances solved w/o backtracking
Number of combinations = O(em)
Size of each combination = m
Twelve combinations for R(*,3)C E R6 R5 R7 R2 R1 R3 B A D C Ry Rx F
R’3 E F R6 R5 R7 R4 R2 R1 R3
R’3 B A D C Ra
UNSAT
SAT
{R1,R4,R6}
{R1,R5,R6}
{R1,R5,R7}
{R1,R6,R7}
{R2,R3,R4}
{R5,R6,R7}
{R1,R2,R3}
{R1,R2,R4}
{R1,R2,R5}
{R1,R2,R6}
{R1,R3,R4}
{R1,R4,R5} B A D C
cl-proj-R(∗,|ψ(cli)|)C
cl-proj-R(∗,|ψ(cli)|)C
cl-proj-wR(∗,3)C
cl-R(∗,|ψ(cli)|)C
..… φ Ri τi τj
… after triangulation
cl-proj-wR(∗,2)C
GAC
cl-proj-wR(∗,3)C
cl-R(∗,|ψ(cli)|)C B A D C
cl-proj-wR(∗,2)C
GAC
Comparing Consistency Properties
Data Characteristics
Localization & projection always outperformed ‘global’
Bolstering schemes of increasing ‘sophistication’ investigate tradeoff between
the effectiveness of constraint filtering &
the cost of generating and maintaining redundant constraints
In tests, projection yields the right tradeoff between the strength of the enforced consistency & the overhead of processing redundant constraints
GAC
maxRPWC
cl+clq-R(∗,3)C
cl+clq-R(∗,4)C
cl+clq-R(∗,2)C
R(∗,4)C
cl-R(∗,|ψ(cli)|)C
cl+proj-R(∗,|ψ(cli)|)C
cl+proj-R(∗,3)C
R(∗,3)C
cl-R(∗,2)C
cl+bin-R(∗,4)C
cl+bin-R(∗,3)C
cl+bin-R(∗,|ψ(cli)|)C
cl+clq-R(∗,|ψ(cli)|)C
cl+bin-R(∗,2)C
cl+proj-R(∗,2)C
R(∗,2)C
cl-R(∗,3)C
cl-R(∗,4)C
cl+proj-R(∗,4)C
max
median
mean
UNSAT
SAT
treewidth
243
158 33 18
43.45
34.44
largest sep.
214
157 28
16.5
39.02
31.33
Max(|ψ(cli)|)
local
1,243
211 16 8
109.54
18.35
projection 11
114.70
37.35
binary
653 24 12
199.50
80.65
clique
148 10
113.35
25.87
clique arity 48 26 7 4
7.40
5.97
Future Research Directions
Automatic selection of consistency property: inside clusters & during search
Modify the structure of a tree decomposition to improve performance (e.g., merging clusters to reduce the overlap [Fattah and Dechter 1996])
Experiments were conducted on the equipment of the Holland Computing Center at the University of Nebraska-Lincoln.
This research is supported by NSF Grant No. RI
July 17th, 2013