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### CS 4700:Foundations of Artificial Intelligence

Carla P. Gomes

Module:

Instance Hardness and Phase Transitions

(Reading R&N: page 224-225)

Beyond NP-Completeness

- NP-Completeness is a worst-case notion!
- Not all problems instances are the same!
- We now have means for discriminating easy from hard instances
- structural differences between instances of the same problem class.

1820

165

150

Are all the Latin Square Instances

(of same size) Equally Difficult?

What is the fundamental difference between instances?

150

Fraction of preassignment:

35%

Are all Latin Square Instances

Equally Difficult?

1820

165

50%

40%

Underconstrained

area

Overconstrained area

20%

42%

50%

Complexity of Latin Square CompletionMedian Runtime (log scale)

Fraction of pre-assignment

Phase transition

from almost all solvable

to almost all unsolvable

Almost all solvable

area

Almost all unsolvable

area

Phase Transition

Fraction of unsolvable cases

Fraction of pre-assignment

Latin Squares with Holes

- Given a full Latin Square, “punch” holes into it

Difficulty:how to generate the full quasigroup, uniformly.

Question:does this give challenging instances?

Markov Chain Monte Carlo (MCMM)

- We use a Markov chain Monte Carlo method (MCMM) whose stationary (ergodic) distribution is uniform over the space of NxN Latin Squares(Jacobson and Matthews 96).
- Startwith arbitrary Latin Square
- Random walk on a sequence of Squares obtained via local modifications

Generation of Latin Squares with Holes (LSWH)

- Use MCMM to generate solved Latin Square
- Punch holes - i.e.,uncolor a fraction of the entries
- The resulting instances are guaranteed satisfiable
- LSWH is NP-Hard

Is there % holes where instances are truly hard on average?

Easy-Hard-Easy Pattern in Backtracking Search

Peak near 32%

(LSCP peaks near 42%)

Computational Cost

Research Question:

why the peak?

% holes

Order 30, 33, 36

Easy-Hard-Easy Pattern in Local SearchResearch Question:

why the peak?

Computational Cost

% holes

First solid statistics for overconstrainted area!

These results for Latin Squares - a structured problem -nicely complement previous results on phase transition and computational complexity for random instances such as SAT, Graph Coloring, etc.

Propositional Satisfiability problem(SAT)

- Satifiability (SAT): Given a formula in propositional logic, is there a model
- (i.e., a satisfying interpretation, an assignment to its variables) making it true?
- We consider clausal form, e.g.:
- ( ab c ) AND ( b c) AND ( ac)

possible assignments

SAT: prototypical hard combinatorial search and reasoning problem. Problem is NP-Complete. (Cook 1971)

Surprising “power” of SAT for encoding computational problems.

(all clauses

have 3 literals)

Early

Results

Complexity peak

Runtime

of interest for alg. design

More about

Propositional

Logic and SAT

in the Logic

module

SAT phase

The phase transition

Prob. satisfiable

UNSAT phase

Thanks

Bart Selman!

Ratioof Clauses to Varables

Random Walk

DP

DP’

GSAT

Walksat

SP

Random 3-SAT as of 2005

Linear time algs.

Mitchell, Selman, and Levesque ’92

5.081

4.762

4.596

4.506

4.601

4.643

Random Walk

DP

DP’

GSAT

Walksat

SP

Random 3-SAT as of 2005

Linear time algs.

Upper bounds

by combinatorial

arguments

(’92 – ’05)

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