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Formal Models of Heavy-Tailed Behavior in Combinatorial Search

Formal Models of Heavy-Tailed Behavior in Combinatorial Search

Hubie Chen, Carla P. Gomes, and Bart Selman

{hubes,gomes,selman}@cs.cornell.edu

Department of Computer Science

Cornell University

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CP - 2001

BackgroundBackground

Randomized backtrack search methods demonstrate high variability of run time

(relative to fixed instance):

Heavy-tailed behavior (Gomes et. al. CP ‘97, JAR ‘00)

New insights into the the design of search algorithms restart strategies

Randomization and restart strategies are now an integral part of state-of-the-art SAT Solvers

(Chaff, GRASP, RELSAT, SATZ-Rand)

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GoalsGoals

Our goals: Formal analysis of tree search models: show

under what conditions heavy-tailed distributions can and cannot arise.

Understand when restart strategies are/are not effective.

Research on heavy-tails in search thus far largely based on empirical studies.

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IntuitionIntuitionIntuitionIntuition

How does heavy-tailed behavior arise?

• The procedure is characterized by a large variability, which leads to highly different trees from run to run.

• Wrong branching decisions may lead the search procedure to explore exponentially large subtrees of the search space containing no solutions.

• A lucky sequence of good branching decisions may lead the search to find a solution after exploring only a small subtree.

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Intuition Pump: RestartsIntuition Pump: RestartsIntuition Pump: RestartsIntuition Pump: Restarts

When are restarts effective?

Suppose a search procedure requires (on inputs of size n):

• Time p(n) (for a polynomial p) with probability ½• Time 2^n with probability ½

No restarts: expected time exponential: equal to ½ * (p(n) + 2^n)

Restart with time interval p(n): expected time drops to polynomial: equal to 2*p(n)

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Outline of TalkOutline of Talk

• Empirical evidence of Heavy-Tailed behavior

• Tree Search Models

• Balanced Tree Search Model

• Imbalanced Tree Search Model

• Bounded Heavy-Tailed Behavior: finite distributions

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Empirical Evidence

of Heavy-Tailed Behavior

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Quasigroups or Latin Squares:An Abstraction for Real World Applications

Quasigroups or Latin Squares:An Abstraction for Real World Applications

Quasigroup or Latin Square

(Order 4)

32% preassignment

Gomes and Selman 96

A quasigroup is an n-by-n matrix such that each row and column is a

permutation of the same n colors

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Randomized Backtrack SearchRandomized Backtrack Search

(*) no solution found - reached cutoff: 2000

Time: (*)3011 (*)7

Easy instance – 15 % preassigned cells

Gomes et al. 97

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Median = 1!

samplemean

3500!

Erratic Behavior of Search CostQuasigroup Completion ProblemErratic Behavior of Search Cost

Quasigroup Completion Problem

500

2000

number of runs

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Heavy-Tailed Distributions

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Heavy-Tailed DistributionsHeavy-Tailed Distributions

• Infinite variance, infinite mean

• Introduced by Pareto in the 1920’s --- “probabilistic curiosity.”

• Mandelbrot established the use of heavy-tailed distributions to model real-world fractal phenomena.

• Examples: stock-market, earthquakes, weather, web traffic...

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Decay of DistributionsDecay of Distributions

Standard

Exponential Decay

e.g. Normal:

Heavy-Tailed

Power Law Decay

e.g. Pareto-Levy:

0,]Pr[ 2

CsomeforxCexX

Pr[ ] ,X x Cx x 0

Power Law Decay

Standard Distribution(finite mean & variance)

Exponential Decay

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Visualization of Heavy Tailed Behavior

Visualization of Heavy Tailed Behavior

Log-log plot of tail of distributionshould be approximately linear.

Slope gives value of

infinite mean and infinite infinite mean and infinite variancevariance

infinite varianceinfinite variance

1

21

466.0

319.0

153.0

Number backtracks (log)

(1-F

(x))

(log

)

Un

solv

ed f

ract

ion

1 => Infinite mean

18% unsolved

0.002% unsolved

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Exploiting Heavy-Tailed BehaviorExploiting Heavy-Tailed Behavior

Heavy Tailed behavior has been observed in several domains: QCP, Graph Coloring, Planning, Scheduling, Circuit synthesis, Decoding, etc.

Consequence for algorithm design:

Use restarts or parallel / interleaved runs to

exploit the extreme variance performance.

Restarts provably eliminate heavy-tailed behavior (Gomes et al. 2000)

70%unsolved

1-F

(x)

Un

solv

ed f

ract

ion

Number backtracks (log)

250 (62 restarts)

0.001%unsolved

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Tree Search Models:

Balanced Tree Model

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Balanced Tree Model, DescribedBalanced Tree Model, DescribedBalanced Tree Model, DescribedBalanced Tree Model, Described

Trees All leaves occur at the same depth Branching factor 2 Exactly one “satisfying” leaf

Search algorithm Chronological backtrack search model Random child selection with no propagation mechanisms

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Balanced Tree Model: AnalysisBalanced Tree Model: AnalysisBalanced Tree Model: AnalysisBalanced Tree Model: Analysis

Let denote the runtime: number of leaf nodes visited (including “satisfying” leaf), on tree of depth n.

Let denote choice at (unique) node above satisfying leaf at depth i :

1 = bad choice, 0 = good choiceThen,

There is exactly one choice of zero-one assignments to the variables for each possible value of T(n); any such assignment has probability

T(n) has an uniform distribution.ni

ninTP 2,,1,

21])([

1022121

)( nXiniXnXnT

n

21

T=4

T=64

iX

)(nT

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Balanced Tree Model: Balanced Tree Model: DistributionDistribution

Balanced Tree Model: Balanced Tree Model: DistributionDistribution

• The expected run time and variance scale exponentially, in the height of the search tree (number of variables);

• The run time distribution is uniform --

shape not heavy tailed.

221)]([

nnTE

12

122)]([n

nTV

(see paper for formal proofs)

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Balanced Tree Model: RestartsBalanced Tree Model: RestartsBalanced Tree Model: RestartsBalanced Tree Model: RestartsRestart strategies are not effective for this model:

no restart strategy with expected polynomial time.

Define a restart strategy to be a sequence of times

Applied to a search procedure by running procedure for time ; restarting and running for time , etc., until

solution found.

Luby et al. (IPL ‘93) show that optimal performance (minimum expectation) obtained by a purely uniform restart strategy:

),...(3),(2),(1 ntntnt

...)(3)(2)(1 ntntnt

)(2 nt)(1 nt

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What sort of improvements can be made to an algorithm so that behavior not like backtrack in balanced tree model?

Very clever search heuristics that lead quickly to the solution node - but that is hard in general

Combination of pruning, propagation, dynamic variable ordering: prune subtrees that do not contain the solution, allowing for runs that are short.

Resulting trees may vary dramatically from run to run.

Balanced Tree ModelBalanced Tree ModelBalanced Tree ModelBalanced Tree Model

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Tree Search Models:

Imbalanced Tree Model

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Imbalanced Tree ModelImbalanced Tree ModelImbalanced Tree ModelImbalanced Tree Model

Algorithm requires time b^iwith probability (1-p)p^i

Intuition: lower p corresponds to “smarter” search

Let T denote the runtime of the algorithm:

the number of leaf nodes visited up to and including the successful

node.

)0()1(][ iippibTP

b=2

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Imbalanced Tree ModelImbalanced Tree Model

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Imbalanced Tree Model:Three Regimes of Behavior

Imbalanced Tree Model:Three Regimes of Behavior

Regime 1:finite expected time, finite variance

Regime 2:finite expected time, infinite variance

Regime 3:infinite expected time, infinite variance

Tail:

when we have

bp 1

LCp

bLpLTP log

2][

(see paper for formal proofs)

21b

p

bp

b1

21

21b

p 2

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Bounded Imbalanced Tree Model

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Bounded Imbalanced Tree Model

Bounded Imbalanced Tree Model

0)1(][ iippibTPUnbounded model Single infinite distribution.

11

0

)1(

npn

i

ipp

Bounded model Infinite number of distributions, one for each n.Arises from truncating successively larger finite segments of unbounded distribution.

Given that:

niipnCn

p

ippibTP ,,1,01

1

)1(][

11

1

np

pnC

We define:

with

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Bounded Imbalanced Tree Model: Three Regimes of Behavior

Bounded Imbalanced Tree Model: Three Regimes of Behavior

Regime 1:

polynomial expected time, polynomial variance

Regime 2:

polynomial expected time, exponential variance

Regime 3:

exponential expected time, exponential variance

bp 1

(see paper for formal proofs)

21b

p

bp

b1

21

Restart strategy - Expected polynomial time

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Bounded Heavy-Tailed BehaviorBounded Heavy-Tailed Behavior

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Balanced, Unbounded, and Imbalanced Trees

Balanced, Unbounded, and Imbalanced Trees

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Conclusions

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ConclusionsConclusions

Heavy-tailed behavior yields insight into backtrack search methods, providing an explanation for the effectiveness of restart strategies.

Tree Search Models: can be analyzed rigorously.

• Balanced Tree Search Model Uniform distribution (not heavy-tailed); restarts are not effective

• Imbalanced Tree Search Model (Bounded/Unbounded) Heavy-tailed; restarts are effective

Consequence for algorithm design: aim for strategies which have highly asymmetric distributions.

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www.cs.cornell.edu/hubeswww.cs.cornell.edu/gomes

Check also:

www.cis.cornell.edu/iisi

www.cs.cornell.edu/hubeswww.cs.cornell.edu/gomes

Check also:

www.cis.cornell.edu/iisi

Demos, papers, etc.