Generating Satisfiable Problem Instances
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Generating Satisfiable Problem Instances Dimitris Achlioptas Microsoft Carla P. Gomes Cornell University Henry Kautz University of Washington Bart Selman Cornell University. AAAI00 Austin, Texas. Introduction.

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Generating Satisfiable Problem InstancesDimitris AchlioptasMicrosoftCarla P. Gomes Cornell University Henry KautzUniversity of WashingtonBart SelmanCornell University

  • AAAI00

  • Austin, Texas


Introduction
Introduction

  • 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 unsatisfiableeasy 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 incompletelocal 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” Formulas

    Problem: the resulting instances are easy – have many satisfying assignments


Outline
Outline

  • 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


Quasigroup or Latin Squares

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 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

32% holes

Quasigroup with Holes(QWH)

  • Given a full quasigroup, “punch” holes into it

Difficulty: how to generate the full quasigroup, uniformly.

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)

  • Use MCMM to generate solved Latin Square

  • 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

Complete (Satz) Search

Order 30, 33, 36

Easy-Hard-Easy Pattern in Backtracking Search

QWH peaks near 32%

(QCP peaks near 42%)

Computational Cost

% holes


Easy hard easy pattern in local search

Local (Walksat) Search

Order 30, 33, 36

Easy-Hard-Easy Pattern in Local Search

Computational Cost

% holes

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

Backbone

Backbone is the shared structure of all

solutions to a given instance (not counting preassigned cells)

Number sols = 2

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!


Sudden phase Transition in Backbone

and it coincides with the hardest area

New Phase Transition in Backbone

% Backbone

% of Backbone

Computational

cost

% holes


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 hardness1
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 backbone
Reparameterization of Backbone

Backbone for different orders (30 - 57)

% of Backbone


Reparameterization computational cost
ReparameterizationComputational Cost

Computational Cost different orders (30, 33, 36)

% of Backbone

Local Search

(normalized)

Local Search

(normalized & reparameterized)


Summary
Summary

  • 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/kautzSATLIBCSPLIB


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