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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
  • 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

  • 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
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?


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% holesQuasigroup 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?


Preassigned cells


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



% 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


Local Search

(normalized & reparameterized)

  • 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)