balance and filtering in structured satisfiability problems
DESCRIPTION
Balance and Filtering in Structured Satisfiability Problems. Henry Kautz University of Washington joint work with Yongshao Ruan (UW), Dimitris Achlioptas (MSR), Carla Gomes (Cornell), Bart Selman (Cornell), Mark Stickel (SRI) CORE – UW, MSR, Cornell. Speedup Learning. - PowerPoint PPT PresentationTRANSCRIPT
Balance and Filtering in Structured Satisfiability Problems
Henry Kautz
University of Washingtonjoint work with
Yongshao Ruan (UW), Dimitris Achlioptas (MSR), Carla Gomes (Cornell), Bart Selman (Cornell),
Mark Stickel (SRI)
CORE – UW, MSR, Cornell
Speedup Learning Machine learning historically considered
Learning to classify objects Learning to search or reason more efficiently
Speedup Learning Speedup learning disappeared in mid-90’s
Last workshop in 1993 Last thesis 1998
What happened? EBL (without generalization) “solved”
rel_sat (Bayardo), GRASP (Silva 1998), Chaff (Malik 2001) – 1,000,000 variable verification problems
EBG too hard algorithmic advances outpaced any successes
Alternative Path
Predictive control of search and reasoning Learn statistical model of behavior of a problem solver
on a problem distribution Use the model as part of a control strategy to improve
the future performance of the solver Synthesis of ideas from
Phase transition phenomena in problem distributions Decision-theoretic control of reasoning Bayesian modeling
Big Picture
ProblemInstances
Solver
static features
runtime
Learning /Analysis
PredictiveModel
dynamic features
resource allocation / reformulation
control / policy
Case Study: Beyond 4.25
ProblemInstances
Solver
static features
runtime
Learning /Analysis
PredictiveModel
Phase transitions & problem hardness
Large and growing literature on random problem distributions
Peak in problem hardness associated with critical value of some underlying parameter
3-SAT: clause/variable ratio = 4.25 Using measured parameter to predict hardness of
a particular instance problematic! Random distribution must be a good model of actual
domain of concern Recent progress on more realistic random
distributions...
Quasigroup Completion Problem (QCP)
NP-Complete Has structure is similar to that of real-world problems -
tournament scheduling, classroom assignment, fiber optic routing, experiment design, ...
Start with empty grad, place colors randomly Generates mix of sat and unsat instances
Phase Transition
Almost all unsolvable area
Fraction of pre-assignment
Fra
ctio
n o
f u
nso
lvab
le c
ases
Almost all solvable area
Complexity Graph
Phase transition
42% 50%20%
42% 50%20%
Underconstrained area
Critically constrained area
Overconstrained area
Quasigroup With Holes (QWH)
Start with solved problem, then punch holes Generates only SAT instances
Can use to test incomplete solvers Hardness peak at phase transition in size of
backbone (Achlioptas, Gomes, & Kautz 2000)
Easy-Hard-Easy pattern in local search
% holes
Co
mp
uta
tio
na
l Co
st
WalksatOrder 30, 33, 36
“Over” constrained area
Underconstrained area
Are we ready to predict run times?
Problem: high variance
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
0.2 0.25 0.3 0.35 0.4 0.45 0.5
log scale
Deep structural features
Rectangular Pattern(Hall 1945)
Aligned Patternnew result!
Balanced Pattern
Tractable Very hard
Hardness is also controlled by structure of constraints, not just the fraction of holes
Random versus balanced
0.E+00
1.E+07
2.E+07
3.E+07
4.E+07
5.E+07
6.E+07
7.E+07
0.2 0.25 0.3 0.35 0.4 0.45 0.5
Balanced
Random
Random vs. balanced (log scale)
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
0.2 0.25 0.3 0.35 0.4 0.45 0.5
Balanced
Random
Morphing balanced and random
Mixed Model - Walksat
0
10
20
30
40
50
60
70
80
90
100
0.00% 20.00% 40.00% 60.00% 80.00% 100.00%
Percent random holes
Tim
e (s
econds)
order 33
Considering variance in hole pattern
Mixed Model - Walksat
0
10
20
30
40
50
60
70
80
90
100
0 2 4 6 8
variance in # holes / row
tim
e
order 33
Time on log scale
Mixed Model - Walksat
1
10
100
0 2 4 6 8
variance in # holes / row
tim
e (s
econds)
log s
cale
order 33
Balanced patterns yield (on average) problems that are 2 orders of magnitude harder than random patterns
Expected run time decreases exponentially with variance in # holes per row or column
Same pattern (differ constants) for DPPL! At extreme of high variance (aligned model) can
prove no hard problems exist
Effect of balance on hardness
2
( ) kE T C
0.01
0.1
1
10
0 50 100 150 200 250 300
variance in # holes
tim
e (
se
co
nd
s)
Morphing random and rectangular
order 33
artifact of walksatartifact of walksat
Morphing Balanced Random Rectangular
0.1
1
10
100
0 5 10 15 20
variance
tim
e (
se
co
nd
s)
order 33
Intuitions
In unbalanced problems it is easier to identify most critically constrained variables, and set them correctly
Backbone variables
Are we done?
Not yet... Observation 1: While few unbalanced problems
are hard, quite a few balanced problems are easy To do: find additional structural features that
predict hardness Introspection Machine learning (Horvitz et al. UAI 2001) Ultimate goal: accurate, inexpensive prediction of
hardness of real-world problems
Are we done?
Not yet… Observation 2: Significant differences in the SAT
instances in hardest regions for the QCP and QWH generators
QWH
QCP(sat only)
Biases in Generators
An unbiased SAT-only generator would sample uniformly at random from the space of all SAT CSP problems
Practical CSP generators Incremental arc-consistency introduces dependencies Hard to formally model the distribution
QWH generator Clean formal model Slightly biased toward problems with many solutions Adding balance makes small, hard problems
balanced QCP balanced QWH
random QCP random QWH
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
0.2 0.25 0.3 0.35 0.4 0.45 0.5
% holes
flip
s
balanced QCP balanced QWH
random QCP random QWH
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
0.2 0.25 0.3 0.35 0.4 0.45 0.5
% holes
flip
s
balanced QCP balanced QWH
random QCP random QWH
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
0.2 0.25 0.3 0.35 0.4 0.45 0.5
% holes
flip
s
balanced QCP balanced QWH
random QCP random QWH
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
0.2 0.25 0.3 0.35 0.4 0.45 0.5
% holes
flip
s
Conclusions
One small part of an exciting direction for improving power of search and reasoning algorithms
Hardness prediction can be used to control solver policy
Noise level (Patterson & Kautz 2001) Restarts (Horvitz et al (CORE team ) UAI 2001)
Lots of opportunities for cross-disciplinary work Theory Machine learning Experimental AI and OR Reasoning under uncertainty Statistical physics