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