CSP

SAT Workshop, June 2011
IBM Watson Research Center, 2011
©
2011
IBM Corporation
A General
Nogood

Learning Framework
for Pseudo

Boolean Multi

Valued SAT*
Siddhartha Jain
Brown University
Ashish Sabharwal
IBM Watson
Meinolf Sellmann
IBM Watson
* to appear at AAAI

2011
IBM Research
©
2011
IBM Corporation
SAT and CSP/CP Solvers
[complete search]
CP Solvers
:
–
work at a high level: multi

valued variables, linear inequalities, element
constraints, global constraint (knapsack,
alldiff
, …)
–
have access to problem structure and specialized inference algorithms!
SAT
:
–
p
roblem crushed down to one, fixed, simple input format:
CNF, Boolean variables, clauses
–
very simple inference at each node: unit propagation
–
yet, translating CSPs to SAT can be extremely powerful!
E.g., SUGAR (winner of CSP Competitions 2008 & 2009)
How do SAT solvers even come close to competing with CP?
A key contributor: efficient and powerful “
nogood
learning
”
2
June 18, 2011 A General Nogood

Learning Framework
IBM Research
©
2011
IBM Corporation
3
June 18, 2011 A General Nogood

Learning Framework
SAT Solvers as Search Engines
Systematic SAT solvers have become
really efficient at searching fast and learning from “mistakes”
E.g., on an IBM model checking instance from SAT Race 2006, with
~170k variables, 725k clauses
, solvers such as
MiniSat
and
RSat
roughly:
–
Make
2000

5000
decisions
/second
–
Deduce
600

1000
conflicts
/second
–
Learn
600

1000
clauses
/second
(#clauses grows rapidly)
–
Restart
every 1

2 seconds
(aggressive restarts)
IBM Research
©
2011
IBM Corporation
An Interesting Line of Work: SAT

X Hybrids
Goal: try to bring more structure to SAT and/or bring SAT

style
techniques to CSP solvers
Examples:
–
Pseudo

Boolean solvers
: inequalities over binary variables
–
SAT Module Theories (SMT)
: “attach” a theory T to a SAT solver
–
Lazy clause generation
: record clausal reasons for domain changes in a
CP solver
–
Multi

valued SAT
: incorporate multi

valued semantics into SAT
preserve the benefits of SAT
solvers in a more general context
strengthen unit propagation
: unit implication
variables
(UIV)
strengthen
nogood

learning
!
[Jain

O’Mahony

Sellmann, CP

2010]
4
June 18, 2011 A General Nogood

Learning Framework
Starting point for this work
IBM Research
©
2011
IBM Corporation
Conflict Analysis: Example
Consider a CSP with
5
variables:
X
1
,
X
2
,
X
4
{
1
,
2
,
3
}
X
3
{
1
, …,
7
}
X
5
{
1
,
2
}
Pure SAT encoding
: variables
x
11
,
x
12
,
x
13
,
x
21
,
x
22
,
x
23
,
x
31
, …,
x
37
,
x
51
,
x
52
What happens when we set
X
1
=
1
,
X
2
=
2
,
X
3
=
1
?
5
June
18
,
2011
A General Nogood

Learning Framework
x
11
= true
x
22
= true
x
31
= true
x
51
= true
C5
No more propagation,
n
o conflict… really?
IBM Research
©
2011
IBM Corporation
Conflict Analysis: Incorporating UIVs
Unit implication variable (
UIV
)
:
a multi

valued clause is “unit”
as soon as all unassigned “literals”
in it regard the same MV variable
SAT encoding, stronger propagation using UIVs:
6
June 18, 2011 A General Nogood

Learning Framework
x
11
= true
x
22
= true
x
31
= true
x
51
= true
x
41
≠ true
x
32
≠ true
C
5
C2
C
1
x
4
3
= true
C
4
x
42
= true
C3
conflict
IBM Research
©
2011
IBM Corporation
What Shall We Learn?
Not a good idea to set
x
41
≠ true and
x
31
= true
learn the
nogood
(
x
41
= true 
x
31
= false)
Problem?
When we backtrack and set, say,
X
3
to anything in {2, 3, 4, 5},
we end up performing exactly the same analysis again and again!
Solution
: represent conflict graph nodes as
variable
inequations
only
7
June
18
,
2011
A General Nogood

Learning Framework
x
11
= true
x
22
= true
x
31
= true
x
51
= true
x
41
≠ true
x
32
≠ true
C
5
C
2
C1
x
4
3
= true
C
4
x
42
= true
C3
conflict
IBM Research
©
2011
IBM Corporation
CMVSAT

1
Use
UIV
rather than UIP
Use
variable
inequations
as the
core
representation
:
X
1
=
1
,
X
2
=
2
,
X
3
=
1
represented as
X
1
≠
2
,
X
1
≠
3
, …
finer granularity
reasoning!
X
3
doesn’t necessarily need to be
1
for a conflict, it just cannot be
6
or
7
Stronger learned multi

valued clause
:
(
X
4
=
1

X
3
=
6

X
3
=
7
)
–
Upon backtracking, we immediately
infer
X
3
≠
1
,
2
,
3
,
4
,
5
!
8
June 18, 2011 A General Nogood

Learning Framework
IBM Research
©
2011
IBM Corporation
This Work: Generalize This Framework
C
ore representation in implication graph
–
CMVSAT

1
:
variable
inequations
(
X
4
≠
1
)
–
i
n general:
primitive constraints
of the solver
–
example:
linear inequalities (
Y
2
≤
5
,
Y
3
≥
1
)
Constraints
–
CMVSAT

1
:
multi

valued clauses (
X
4
=
1

X
3
=
6

X
3
=
7
)
–
in general:
secondary constraints
supported by the solver
–
example:
(
X
1
= true 
Y
3
≤
4

X
3
= Michigan)
Propagation
of a secondary constraint C
s
:
entailment of a new primitive constraint
from C
s
and known primitives
9
June
18
,
2011
A General Nogood

Learning Framework
IBM Research
©
2011
IBM Corporation
This Work: Generalize This Framework
Learned
nogoods
–
CMVSAT

1:
multi

valued clauses (
X
4
= 1 
X
3
= 6 
X
3
= 7)
–
i
n general:
disjunctions of negations of primitives
(with certain desirable properties)
–
example:
(
X
1
= true 
Y
3
≤ 4 
X
3
= Michigan)
Desirable properties of
nogoods
?
–
The “unit” part of the learned
nogood
that is implied upon
backtracking
must be
representable as a set of primitives
!
–
E.g., if
Y
3
is meant to become unit upon backtracking, then
(
X
1
= true 
Y
3
≤ 4 
Y
3
≥ 6

X
3
= Michigan) is NOT desirable
•
cannot represent
Y
3
≤ 4 
Y
3
≥ 6
as a conjunction of primitives
•
upon backtracking, cannot propagate the learned
nogood
10
June
18
,
2011
A General Nogood

Learning Framework
IBM Research
©
2011
IBM Corporation
Sufficient Conditions
1.
System distinguishes between primitive and secondary constraints
2.
Secondary constraint propagators:
–
Entail new primitive constraints
–
Efficiently provide a set of primitives constraints sufficient for the entailment
3.
Can efficiently detect conflicting sets of primitives
–
and represent the disjunction of their negations (the “
nogood
”)
as a secondary constraint
4.
Certain sets of negated primitives (e.g., those arising from propagation
upon backtracking) succinctly representable as a set of primitives
5.
Branching executed as the addition of one or more* primitives
Under these conditions, we can efficiently learn strong
nogoods
!
11
June 18, 2011 A General Nogood

Learning Framework
[details in AAAI

2011
paper]
IBM Research
©
2011
IBM Corporation
Abstract Implication Graph
(under sufficient conditions)
12
June
18
,
2011
A General Nogood

Learning Framework
C
p
: primary constraints
branched upon or entailed at various decision levels
Learned
nogood
: disjunction of negations of primitives in
shaded
nodes
IBM Research
©
2011
IBM Corporation
General Framework: Example
X
1
{true, false}
X
3
{NY, TX, FL, CA}
X
5
{r, g, b}
X
2
,
X
4
,
X
6
,
X
7
{
1
, …,
100
}
Branch on
X
1
≠ true,
X
2
≤
50
:
Learn:
(
X
3
= FL 
X
4
≥
31
)
13
June
18
,
2011
A General Nogood

Learning Framework
Notes:
1.
T
he part of the
nogood
that is unit upon backtracking
need not always regard the same variable
! (e.g., when
X
≤
Y
is primitive)
2.
Neither is regarding the same variable sufficient
for being a valid
nogood
!
(e.g.,
X
≤
4

X
≥
11
wouldn’t be representable)
IBM Research
©
2011
IBM Corporation
Empirical Evaluation
Ideas implemented in CMVSAT

2
Currently supported:
–
usual
domain
variables
, and
range
variables
–
linear inequalities
, e.g.
(
X
1
+
5
X
2
–
13
X
3
≤
6
)
–
disjunctions
of equations and range constraints
e.g.
(
X
1
[
5
…
60
] 
X
2
[
㌷
…
74
] 
X
5
= b 
X
10
≤
15
)
Comparison against:
–
SAT solver:
Minisat
[
Een

Sorensson
2004
, version
2.2.0
]
–
CSP solver:
Mistral
[
Hebrard
2008
]
–
MIP solver:
SCIP
[
Achterberg
2004
, version
2.0.1
]
Encodings generated using
Numberjack
[
Hebrard
et al
2010
, version
0.1.10

11

24
]
14
June
18
,
2011
A General Nogood

Learning Framework
IBM Research
©
2011
IBM Corporation
Empirical Evaluation
Benchmark domains
(
100
instances of each)
–
QWH

C
: weighted quasi

group with holes / Latin square completion
•
r
andom cost
c
ik
{
1
, …,
10
} assigned to cell (
i
,
k
)
•
c
ost constraint:
sum
ik
(
c
ik
X
ik
) ≤ (sum of all costs) /
2
•
s
ize
25
x
25
,
40
% filled entries, all satisfiable
–
MSP

3
: market split problem
•
n
otoriously hard for constraint solvers
•
p
artition
20
weighted items into
3
equal sets,
10
% satisfiable
–
NQUEENS

W
: weighted n

queens,
30
x
30
•
average weight of occupied cells ≥
70
% of
weight
max
•
size
30
x
30
, random weights
{
1
,…,
10
}
15
June
18
,
2011
A General Nogood

Learning Framework
IBM Research
©
2011
IBM Corporation
Empirical Evaluation: Results
CMVSAT

2
shows good performance across a variety of domains
–
Solved all
300
instances in <
4
sec on average
MiniSat
not suitable for domains like QWH

C
–
Encoding (even “compact” ones like in Sugar) too large to generate or solve
–
20
x slower on MSP

3
Mistral
explores a much larger search space than needed
–
Lack of
nogood
learning becomes a bottleneck:
e.g.:
3
M nodes
for QWH

C
(
36
% solved)
, compared to
231
nodes
for CMVSAT

2
SCIP
takes
100
x longer on QWH

c,
10
x longer on NQUEENS

W
16
June
18
,
2011
A General Nogood

Learning Framework
SAT solver
CSP solver
MIP solver
IBM Research
©
2011
IBM Corporation
Summary
A
generalized
framework for
SAT

style
nogood

learning
–
extends CMVSAT

1
–
low

overhead process, retains efficiency of SAT solvers
–
sufficient conditions
:
•
primitive constraints
•
s
econdary constraints, propagation as entailment of primitives
•
valid
cutsets
in conflict graphs: facilitate propagation upon backtracking
•
other efficiency / representation criteria
CMVSAT

2
:
robust performance across a variety of problem domains
–
compared to a SAT solver, a CSP solver, a MIP solver
–
open: more extensive evaluation and comparison against, e.g., lazy clause
generation, pseudo

Boolean solvers, SMT solvers
–
nonetheless,
a promising and fruitful direction!
17
June
18
,
2011
A General Nogood

Learning Framework
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