AAAI00 Austin, Texas Generating Satisfiable Problem Instances Dimitris Achlioptas Microsoft Carla P. Gomes Cornell University Henry Kautz University of Washington Bart Selman Cornell University
Jan 08, 2016
AAAI00
Austin, Texas
Generating Satisfiable Problem Instances
Dimitris AchlioptasMicrosoft
Carla P. Gomes Cornell University
Henry KautzUniversity of Washington
Bart SelmanCornell University
Generating Satisfiable Problem Instances
Dimitris AchlioptasMicrosoft
Carla P. Gomes Cornell University
Henry KautzUniversity of Washington
Bart SelmanCornell University
IntroductionIntroduction
An important factor in the development of search methods is the availability of good benchmarks.
Sources for benchmarks:
• Real world instances– hard to find
– too specific
• Random generators– easier to control (size/hardness)
Random Generators of Instances
Random Generators of Instances
Understanding threshhold phenomena lets us tune the hardness of problem instances:•At low ratios of constraints -
• most satisfiable, easy to find assignments;
•At high ratios of constraints -
• most unsatisfiable easy to show inconsistency;
•At the phase transition between these two regions
• roughly half of the instances are satisfiable and we find a concentration of computationally hard instances.
Limitation of Random Generators
Limitation of Random Generators
PROBLEM: evaluating incomplete local search algorithms
•Filtering out Unsat Instances - use a complete method and throw away unsat instances.
Problem: want to test on instances too large for any complete method!
•“Forced” FormulasProblem: the resulting instances are easy – have
many satisfying assignments
OutlineOutline
I Generation of only satisfiable instances
II New phase transition in the space of satisfiable instances
III Connection between hardness of satisfiable instances and new phase transition
IV Conclusions
Generation of only satisfiable instances
Generation of only satisfiable instances
Given an N X N matrix, and given N colors, color the matrix in such a way that:
-all cells are colored;
- each color occurs exactly once in each row;
- each color occurs exactly once in each column;
Quasigroup or Latin Square
Quasigroup or Latin Squares
Quasigroup Completion Problem (QCP)
Quasigroup Completion Problem (QCP)
Given a partial assignment of colors (10 colors in this case), can the partial quasigroup (latin square) be completed so we obtain a full quasigroup?
Example:
32% preassignment
QCP: A Framework for Studying Search
QCP: A Framework for Studying Search
•NP-Complete.
•Random instances have structure not found in random k-SAT
Closer to “real world” problems!
•Can control hardness via % preassignment
•BUT problem of creating large, guaranteed satisfiable instances remains…
(Anderson 85, Colbourn 83, 84, Denes & Keedwell 94, Fujita et al. 93, Gent et al. 99, Gomes & Selman 97, Gomes et al. 98, Shaw et al. 98, Walsh 99 )
Quasigroup with Holes(QWH)
Quasigroup with Holes(QWH)
Given a full quasigroup, “punch” holes into it
Difficulty: how to generate the full quasigroup, uniformly.32% holes
Question: does this give challenging instances?
Markov Chain Monte Carlo (MCMM)
Markov Chain Monte Carlo (MCMM)
We use a Markov chain Monte Carlo method (MCMM) whose stationary (egodic) distribution is uniform over the space of NxN quasigroups (Jacobson and Matthews 96).
• Start with arbitrary Latin Square
• Random walk on a sequence of Squares obtained via local modifications
Generation of Quasigroup with Holes (QWH)
Generation of Quasigroup with Holes (QWH)
1) Use MCMM to generate solved Latin Square
2) Punch holes - i.e., uncolor a fraction of the entries
• The resulting instances are guaranteed satisfiable
• QWH is NP-Hard
Is there % holes where instances truly hard on average?
Easy-Hard-Easy Pattern in Backtracking Search
Easy-Hard-Easy Pattern in Backtracking Search
% holes
Co
mp
uta
tio
na
l Co
stComplete (Satz) Search
Order 30, 33, 36
QWH peaks near 32%
(QCP peaks near 42%)
Easy-Hard-Easy Pattern in Local Search
Easy-Hard-Easy Pattern in Local Search
% holes
Co
mp
uta
tio
na
l Co
st
Local (Walksat) SearchOrder 30, 33, 36
First solid statistics for overconstrainted area!
Phase Transition in QWH?Phase Transition in QWH?
QWH - all instances are satisfiable - does it still make sense to talk about a phase transition?
• The standard phase transition corresponds to the area with 50% SAT/UNSAT instances
• Here all instances SAT
Does some other property of the wffs show an abrupt change around “hard” region?
Backbone
Preassigned cells
Number sols = 4
Backbone
Backbone is the shared structure of allsolutions to a given instance (not counting preassigned cells)
Backbone size = 2
Phase Transition in the Backbone
Phase Transition in the Backbone
We have observed a transition in the size of backbone
• Many holes – backbone close to 0%• Fewer holes – backbone close to
100%• Abrupt transition – coincides with
hardest instances!
New Phase Transition in Backbone
% Backbone
Sudden phase Transition in Backboneand it coincides with the hardest area
% holes
Computationalcost%
of
Bac
kbo
ne
Why correlation between backbone and problem hardness?
Why correlation between backbone and problem hardness?
Intuitions: Local Search
•Near 0% Backbone = many solutions = easy to find by chance
•Near 100% Backbone = solutions tightly clustered = all the constraints “vote” in same direction
•50% Backbone = solutions in different clusters = different clauses push search toward different clusters
(Current work – verify intuitions!)
Why correlation between backbone and problem hardness?
Why correlation between backbone and problem hardness?
Intuitions: Backtracking search
Bad assignments to backbone variables near root of search tree cause the algorithm to deteriorate
For the algorithm to have a significant chance of making bad choices, a non-negligible fraction of variables must appear in the backbone
Reparameterization of BackboneReparameterization of Backbone%
of
Bac
kbo
ne
Backbone for different orders (30 - 57)
)2/( NHolesNum )55.1/( NHolesNum
ReparameterizationComputational CostReparameterizationComputational Cost
Computational Cost different orders (30, 33, 36)
)55.1/( NHolesNum
% o
f B
ackb
on
e
Local Search (normalized)
)55.1/( NHolesNum
Local Search (normalized & reparameterized)
SummarySummary
QWH is a problem generator for satisfiable instances (only):
• Easy to tune hardness• Exhibits more realistic structure • Well-suited for the study of incomplete search
methods (as well as complete)• Confirmation of easy-hard-easy pattern in
computational cost for local search
New kind of phase transition in backbone• Reparameterization
GOAL – new insights into practical complexity of problem solving
QWH generator, demos, available soon (< one month):
www.cs.cornell.edu/gomeswww.cs.washington.edu/home/kautz
SATLIBCSPLIB
QWH generator, demos, available soon (< one month):
www.cs.cornell.edu/gomeswww.cs.washington.edu/home/kautz
SATLIBCSPLIB