carla p. gomes school on optimization cpaior02 exploiting structure and randomization in...

Carla P. Gomes

School on Optimization CPAIOR02

Exploiting Structure and Randomization

in Combinatorial Search

Carla P. [email protected]

www.cs.cornell.edu/gomes

Intelligent Information Systems InstituteDepartment of Computer Science

Cornell University

Exploiting Structure and Randomization

in Combinatorial Search

Carla P. [email protected]


Intelligent Information Systems InstituteDepartment of Computer Science

Cornell University

Carla P. Gomes


OutlineOutline

A Structured Benchmark Domain

Randomization

Conclusions

Carla P. Gomes


Given an N X N matrix, and given N colors, a quasigroup of order N is a a colored matrix, such 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(Order 4)

Quasigroups or Latin Squares:An Abstraction for Real World Applications

Carla P. Gomes


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

(Gomes & Selman 97)

Carla P. Gomes


Quasigroup Completion Problem A Framework for Studying SearchQuasigroup Completion Problem

A Framework for Studying Search

NP-Complete.

Has a structure not found in random instances,

such as random K-SAT.

Leads to interesting search problems when structure is perturbed (more about it later).

Good abstraction for several real world problems: scheduling and timetabling, routing in fiber optics, coding, etc(Anderson 85, Colbourn 83, 84, Denes & Keedwell 94, Fujita et al. 93, Gent et al. 99, Gomes & Selman 97, Gomes et al. 98, Meseguer & Walsh 98, Stergiou and Walsh 99, Shaw et al. 98, Stickel 99, Walsh 99 )

Carla P. Gomes


Fiber Optic Networks

Nodesconnect point to point

fiber optic links

Carla P. Gomes


Fiber Optic Networks

Nodesconnect point to point

fiber optic links

Each fiber optic link supports alarge number of wavelengths

Nodes are capable of photonic switching --dynamic wavelength routing --

which involves the setting of the wavelengths.

Carla P. Gomes


Routing in Fiber Optic Networks

Routing Node

How can we achieve conflict-free routing in each node of the network?

Dynamic wavelength routing is a NP-hard problem.

Input Ports Output Ports1

2

3

4

1

2

3

4

preassigned channels

Carla P. Gomes


QCP Example Use: Routers in Fiber Optic Networks

QCP Example Use: Routers in Fiber Optic Networks

Dynamic wavelength routing in Fiber Optic Networks can be directly mapped into the Quasigroup Completion Problem.

(Barry and Humblet 93, Cheung et al. 90, Green 92, Kumar et al. 99)

•each channel cannot be repeated in the same input port (row constraints);• each channel cannot be repeated in the same output port (column constraints);

CONFLICT FREELATIN ROUTER

Inp

ut

po

rts

Output ports

3

1

2

4

Input Port Output Port

1

2

43

Carla P. Gomes


Traditional View of Hard Problems - Worst Case View

Traditional View of Hard Problems - Worst Case View

“They’re NP-Complete—there’s no way to do anything but try heuristic approaches and hope for the best.”

Carla P. Gomes


New Concepts in ComputationNew Concepts in Computation

Not all NP-Hard problems are the same!

We now have means for discriminating easy from hard instances

---> Phase Transition concepts

Carla P. Gomes


NP-completeness is a worst-case notion – what about average

complexity?

Structural differences between instances of the same NP- complete problem (QCP)

NP-completeness is a worst-case notion – what about average

complexity?

Structural differences between instances of the same NP- complete problem (QCP)

Carla P. Gomes


Are all the Quasigroup Instances(of same size) Equally Difficult?

1820150

Time performance:

165

What is the fundamental difference between instances?

Carla P. Gomes


Are all the Quasigroup Instances

Equally Difficult?

1820 165

40% 50%

150

Time performance:

35%

Fraction of preassignment:

Carla P. Gomes


Complexity of Quasigroup Completion

Complexity of Quasigroup Completion

Fraction of pre-assignment

Med

ian

Ru

nti

me

(log

sca

le)

Critically constrained area

Overconstrained areaUnderconstrained

area

42% 50%20%

Carla P. Gomes


Phase Transition

Almost all unsolvable area

Fraction of pre-assignmentFra

ctio

n o

f u

nso

lvab

le c

ases

Almost all solvable area

Complexity Graph

Phase transition from almost all solvableto almost all unsolvable

Carla P. Gomes


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

(Broder et al. 93; Clearwater and Hogg 96, Cheeseman et al. 91, Cook and Mitchell 98, Crawford and Auton 93, Crawford and Baker 94, Dubois 90, Frank et al. 98, Frost and Dechter 1994, Gent and Walsh 95, Hogg, et al. 96, Mitchell et al. 1992, Kirkpatrick and Selman 94, Monasson et 99, Motwani et al. 1994, Pemberton and Zhang 96, Prosser 96, Schrag and Crawford 96, Selman and Kirkpatrick 97, Smith and Grant 1994, Smith and Dyer 96, Zhang and Korf 96, and more)

Carla P. Gomes


QCPDifferent Representations /

Encodings

QCPDifferent Representations /

Encodings

Carla P. Gomes


Cubic representation of QCP

Columns

Rows

Colors

Carla P. Gomes


QCP as a MIPQCP as a MIP

• Variables -

• Constraints -

}1,0{ijk

x

....,,2,1,,;, nkjikcolorhasjicellijk

x

....,,2,1,,1,

nkjii ijk

xkj

)3(nO

)2(nO

....,,2,1,,1,, nkjik ijk

xji

....,,2,1,,1,

nkjij ijkx

ki

Row/color line

Column/color line

Row/column line

Carla P. Gomes


QCP as a CSPQCP as a CSP

• Variables -

• Constraints -

}...,,2,1{, njix

....,,2,1,;,, njijicellofcolorjix

....,,2,1);,,...,2,

,1,

( ninixix

ixalldiff

....,,2,1);,,...,,2

,,1

( njjnxj

xj

xalldiff

)2(nO

)(nO

row

column

[ vs. for MIP])3(nO

[ vs. for MIP])2(nO

Carla P. Gomes


Exploiting Structure for Domain Reduction

Exploiting Structure for Domain Reduction

• A very successful strategy for domain reduction in CSP is to exploit the structure of groups of constraints and treat them as global constraints.

Example using Network Flow Algorithms:

• All-different constraints

(Caseau and Laburthe 94, Focacci, Lodi, & Milano 99, Nuijten & Aarts 95, Ottososon & Thorsteinsson 00, Refalo 99, Regin 94 )

Carla P. Gomes


Exploiting Structure in QCPALLDIFF as Global Constraint

Two solutions:

we can update the domains of the column

variables

Analogously, we can update the domains of the other variables

Matching on a Bipartite graph

All-different constraint

(Berge 70, Regin 94, Shaw and Walsh 98 )

Carla P. Gomes


Exploiting StructureArc Consistency vs. All Diff

Arc ConsistencySolves up to order 20

Size search space 40020

AllDiffSolves up to order 33

Size searchspace 108933

Carla P. Gomes


Quasigroup as SatisfiabilityQuasigroup as Satisfiability

Two different encodings for SAT:

2D encoding (or minimal encoding);

3D encoding (or full encoding);

Carla P. Gomes


2D Encoding or Minimal Encoding2D Encoding or Minimal Encoding

Variables:

Each variables represents a color assigned to a cell.

Clauses:

• Some color must be assigned to each cell (clause of length n);

• No color is repeated in the same row (sets of negative binary clauses);

• No color is repeated in the same column (sets of negative binary clauses);

3n

)21

( ijnxij

xij

xij

....,,2,1,,;, nkjikcolorhasjicellijk

x

)1

()31

()21

(ink

xki

xki

xki

xki

xki

xik

)1

()31

()21

(njk

xjk

xjk

xjk

xjk

xjk

xjk

)4(nO

}1,0{ijk

x

Carla P. Gomes


3D Encoding or Full Encoding3D Encoding or Full Encoding

This encoding is based on the cubic representation of the quasigroup: each line of the cube contains exactly one true variable;

Variables:

Same as 2D encoding.

Clauses:

• Same as the 2 D encoding plus:

• Each color must appear at least once in each row;

• Each color must appear at least once in each column;

• No two colors are assigned to the same cell;

)4(nO

Carla P. Gomes


Capturing Structure - Performance of SAT Solvers

Capturing Structure - Performance of SAT Solvers

State of the art backtrack and local search and complete SAT solvers using 3D encoding are very competitive with specialized CSP algorithms.

In contrast SAT solvers perform very poorly on 2D encodings (SATZ or SATO);

In contrast local search solvers (Walksat) perform well on 2D encodings;

Carla P. Gomes


SATZ on 2D encoding(Order 20 -28)

SATZ and SATO can only solve up to order 28 when using 2D encoding;When using 3D encoding problems of the same size take only 0 or 1 backtrack and much higher orders can be solved;

1,000,000Order 28

Order 20

Carla P. Gomes


Walksat on 2D and 3D encoding(Order 30-33)

Walksat on 2D and 3D encoding(Order 30-33)

1,000,0002D order 333D order 33

Walksat shows an unsual pattern - the 2D encodings are somewhat easier than the 3D encoding

at the peak and harder in the undereconstrained region;

Carla P. Gomes


Quasigroup - SatisfiabilityQuasigroup - Satisfiability

Encoding the quasigroup using only

Boolean variables in clausal form using

the 3D encoding is very competitive.

Very fast solvers - SATZ, GRASP,

SATO,WALKSAT;

Carla P. Gomes


Structural features of instances provide insights into their hardness namely:

Backbone

Inherent Structure and Balance

Carla P. Gomes


Backbone

This instance has4 solutions:

Backbone

Total number of backbone variables: 2

Backbone is the shared structure of all the solutions to a given instance.

Carla P. Gomes


Phase Transition in the Backbone

Phase Transition in the Backbone

• We have observed a transition in the backbone from a phase where the size of the backbone is around 0% to a phase with backbone of size close to 100%.

• The phase transition in the backbone is sudden and it coincides with the hardest problem instances.

(Achlioptas, Gomes, Kautz, Selman 00, Monasson et al. 99)

Carla P. Gomes


New Phase Transition in BackboneQCP (satisfiable instances only)

% Backbone

Sudden phase transition in Backbone

Fraction of preassigned cells

Computationalcost

% o

f B

ackb

on

e

Carla P. Gomes


Inherent Structure and Balance

Carla P. Gomes


Quasigroup Patterns and Problems Hardness

Quasigroup Patterns and Problems Hardness

Rectangular Pattern Aligned Pattern Balanced Pattern

Tractable Very hard

(Kautz, Ruan, Achlioptas, Gomes, Selman 2001)

Carla P. Gomes


SATZSATZ

Balanced QCP

Rectangular QCP

Aligned QCP

QCP

QWH

Carla P. Gomes


Walksat Walksat

aligned

rectangular

Balanced filtered QCPBalance QWH

QCPQWH

We observe the same ordering in hardness when using Walksat,SATZ, and SATO – Balacing makes instances harder

Carla P. Gomes


Phase Transitions, Backbone, Balance

Phase Transitions, Backbone, Balance

Summary

The understanding of the structural properties of problem instances based on notions such as phase transitions, backbone, and balance provides new insights into the practical complexity of many computational tasks.

Active research area with fruitful interactions between computer science, physics (approaches

from statistical mechanics), and mathematics (combinatorics / random structures).

Carla P. Gomes


OutlineOutline


Randomization

Conclusions

Carla P. Gomes


Randomized Backtrack Search Randomized Backtrack Search ProceduresProcedures

Randomized Backtrack Search Randomized Backtrack Search ProceduresProcedures

Carla P. Gomes


BackgroundBackground

Stochastic strategies have been very successful in the area of local search.

Simulated annealingGenetic algorithmsTabu SearchGsat and variants.

Limitation: inherent incomplete nature of local search methods.

Carla P. Gomes


BackgroundBackground

We want to explore the We want to explore the addition of aaddition of a

stochastic elementstochastic element to a systematic search to a systematic search

procedure procedure without losing completeness.without losing completeness.

Carla P. Gomes


We introduce stochasticity in a backtrack search method, e.g., by randomly breaking ties in variable and/or value selection.

Compare with standard lexicographic tie-breaking.

Randomization

Carla P. Gomes


RandomizationRandomization

At each choice point break ties (variable selection and/or value selection) randomly or:

“Heuristic equivalence” parameter (H) - at every choice point consider as “equally” good H% top choices; randomly select a choice from equally good choices.

Carla P. Gomes


Randomized StrategiesRandomized Strategies

Strategy Variable sel. Value sel.

DD deterministic deterministic

DR deterministic random

RD random deterministic

RR random random

Carla P. Gomes


Quasigroup DemoQuasigroup Demo

Carla P. Gomes


Distributions of Randomized Backtrack Search

Distributions of Randomized Backtrack Search

Key Properties:

I Erratic behavior of mean

II Distributions have “heavy tails”.

Carla P. Gomes


Median = 1!

samplemean

3500!

Erratic Behavior of Search CostQuasigroup Completion ProblemErratic Behavior of Search Cost

Quasigroup Completion Problem

500

2000

number of runs

Carla P. Gomes


1

Carla P. Gomes


75%<=30

Number backtracks Number backtracks

Pro

port

ion o

f ca

ses

Solv

ed

5%>100000

Carla P. Gomes


Heavy-Tailed DistributionsHeavy-Tailed Distributions

… … infinite variance … infinite meaninfinite 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, earth-quakes, weather,...

Carla P. Gomes


Decay of DistributionsDecay of Distributions

Standard --- Exponential Decay

e.g. Normal:

Heavy-Tailed --- Power Law Decay

e.g. Pareto-Levy:

Pr[ ] , ,X x Ce x for some C x 2 0 1

Pr[ ] ,X x Cx x 0

Carla P. Gomes


Standard Distribution(finite mean & variance)

Power Law Decay

Exponential Decay

Carla P. Gomes


Normal, Cauchy, and LevyNormal, Cauchy, and Levy

Normal - Exponential Decay

Cauchy -Power law DecayLevy -Power law Decay

Carla P. Gomes


Tail Probabilities (Standard Normal, Cauchy, Levy)

Tail Probabilities (Standard Normal, Cauchy, Levy)

c Normal Cauchy Levy0 0.5 0.5 11 0.1587 0.25 0.68272 0.0228 0.1476 0.52053 0.001347 0.1024 0.43634 0.00003167 0.078 0.3829

Carla P. Gomes


Example of Heavy Tailed Model(Random Walk)

Example of Heavy Tailed Model(Random Walk)

Random Walk:•Start at position 0

•Toss a fair coin:

• with each head take a step up (+1)

• with each tail take a step down (-1)

X --- number of steps the random walk takes to return to position 0.

Carla P. Gomes


The record of 10,000 tosses of an ideal coin

(Feller)

Zero crossing Long periods without zero crossing

Carla P. Gomes


Random Walk

Heavy-tails vs. Non-Heavy-TailsHeavy-tails vs. Non-Heavy-Tails

Normal(2,1000000)

Normal(2,1)

O,1%>200000

50%

2

Median=2

1-F

(x)

Unso

lved f

ract

ion

X - number of steps the walk takes to return to zero (log scale)

Carla P. Gomes


How to Check for “Heavy Tails”?How to Check for “Heavy Tails”?

Log-Log plot of tail of distribution

should be approximately linear.

Slope gives value of

infinite mean and infinite varianceinfinite mean and infinite variance

infinite varianceinfinite variance

1

1 2

Carla P. Gomes


466.0

319.0153.0

Number backtracks (log)

(1-F

(x))

(log

)U

nso

lved

fra

ctio

n

1 => Infinite mean

Heavy-Tailed Behavior in QCP Domain

18% unsolved

0.002% unsolved

Carla P. Gomes


Formal Models of Heavy-Tailed Behavior in Combinatorial Search

Chen, Gomes, Selman 2001

Carla P. Gomes


MotivationMotivationMotivationMotivation

Research on heavy-tails has been largely based on empirical studies of run time distribution.

Goal: to provide a formal characterization of tree search models and show under what conditions heavy-tailed distributions can arise.

Intuition: Heavy-tailed behavior arises:

• from the fact that wrong branching decisions may lead the procedure to explore an exponentially large subtree of the search space that contains no solutions;

• the procedure is characterized by a large variability in the time to find a solution on different runs, which leads to highly different trees from run to run;

Carla P. Gomes


Balanced vs. ImbalancedBalanced vs. Imbalanced Tree Model Tree Model

Balanced vs. ImbalancedBalanced vs. Imbalanced Tree Model Tree Model

Balanced Tree Model:

• chronological backtrack search model;• fixed variable ordering;• random child selection with no propagation

mechanisms;

(show demo)

Carla P. Gomes


221)]([

nnTE

12

122)]([n

nTV

The run time distribution of chronological backtrack search ona complete balanced tree is uniform (therefore not heavy-tailed).Both the expected run time and variance scale exponentially

Carla P. Gomes


Balanced Tree ModelBalanced Tree ModelBalanced Tree ModelBalanced Tree Model

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

• The run time distribution is Uniform, (not heavy tailed ).

• Backtrack search on balanced tree model has no restart strategy with exponential polynomial time.

221)]([

nnTE

12

122)]([n

nTV

Chen, Gomes & Selman 01

Carla P. Gomes


How can we improve on the balanced serach tree model?

Very clever search heuristic that leads quickly to the solution node - but that is hard in general;

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

---> resulting trees may vary dramatically from run to run.

Carla P. Gomes


T - the number of leaf nodes visited up to and including the successful node; b - branching factor

0)1(][ iippibTP

Formal Model Yielding Heavy-Tailed BehaviorFormal Model Yielding Heavy-Tailed Behavior

b = 2

(show demo)

Carla P. Gomes


Expected Run Time(infinite expected time)

Variance

(infinite variance)

Tail

(heavy-tailed)

][1 TEb

p

][2

1 TVb

p

2log

2][2

1 LCp

bLpLTPb

p

Carla P. Gomes


Bounded Heavy-Tailed BehaviorBounded Heavy-Tailed Behavior

(show demo)

Carla P. Gomes


No Heavy-tailed behavior for Proving Optimality

No Heavy-tailed behavior for Proving Optimality

Carla P. Gomes


Proving OptimalityProving Optimality

Carla P. Gomes


Small-World Vs. Heavy-Tailed Behavior

Small-World Vs. Heavy-Tailed Behavior

Does a Small-World topology (Watts & Strogatz) induce heavy-tail behavior?

The constraint graph of a quasigroup exhibits a small-world topology(Walsh 99)

Carla P. Gomes


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. 97, Hoos 99, Horvitz 99, Huberman, Lukose and Hogg 97, Karp et al 96, Luby et al. 93, Rish et al. 97, Wlash 99)

Carla P. Gomes


X XX XX

solved10 101010 10

Sequential: 50 +1 = 51 seconds

Parallel: 10 machines --- 1 second 51 x speedup

Super-linear Speedups

Interleaved (1 machine): 10 x 1 = 10 seconds 5 x speedup

Carla P. Gomes


RestartsRestarts

70%unsolved

1-F

(x)

Un

solv

ed f

ract

ion


no restarts

restart every 4 backtracks

250 (62 restarts)

0.001%unsolved

Carla P. Gomes


Example of Rapid Restart Speedup(planning)

Example of Rapid Restart Speedup(planning)

1000

10000

100000

1000000

1 10 100 1000 10000 100000 1000000

log( cutoff )

log

( b

ackt

rack

s )

20

2000 ~100 restarts

Cutoff (log)

Num

ber

back

track

s (l

og)

~10 restarts

100000

Carla P. Gomes


Sketch of proof of elimination of heavy tails

Sketch of proof of elimination of heavy tails

Let’s truncate the search procedure after m backtracks.

Probability of solving problem with truncated version:

Run the truncated procedure and restart it repeatedly.

pm X m Pr[ ]

X numberof backtracks to solve the problem

Carla P. Gomes


Y total number backtracks with restarts

F Y y pmY m

c e c y

Pr[ ] ( )

/1

12

Number of starts Y m Geometric pmRe / ~ ( )

Y - does not have Heavy Tails

Carla P. Gomes


Decoding in Communication Systems

Decoding in Communication Systems

Source Encoder Decoder DestinationChannel

Voice waveform, binary digits from a cd, output of a set of sensors in a space probe, etc.

Telephone line, a storage medium, a space communication link, etc.

usually subject to NOISE

Processing prior to transmission,e.g., insertion of redundancy to combat the channel noise. Processing of the channel output with the

objective of producing at the destinationan acceptable replica of the source output.

Decoding in communication systems is NP-hard.

(Berlekamp, McEliece, and van Tilborg 1978, Barg 1998)

Carla P. Gomes


Retransmissions in Sequential Decoding

Retransmissions in Sequential Decoding

1-F

(x)

Un

solv

ed f

ract

ion


without retransmissions

with retransmissions

Gomes et al. 2000 / 20001

Carla P. Gomes


Paramedic Crew AssignmentParamedic Crew Assignment

Paramedic crew assignment is the problem of assigning paramedic crews from different stations to cover a given region, given several resource constraints.

Carla P. Gomes


Deterministic SearchDeterministic Search

Carla P. Gomes


RestartsRestarts

Carla P. Gomes


Deterministic

Logistics Planning 108 mins. 95 sec.

Scheduling 14 411 sec 250 sec

(*) not found after 2 days

Scheduling 16 ---(*) 1.4 hours

Scheduling 18 ---(*) ~18 hrs

Circuit Synthesis 1 ---(*) 165sec.Circuit Synthesis 2 ---(*) 17min.

Results on Effectiveness of Restarts

R3

Carla P. Gomes


Algorithm Portfolio DesignAlgorithm Portfolio Design

Gomes and Selman 1997 - Proc. UAI-97;

Gomes et al 1997 - Proc. CP97.

Carla P. Gomes


MotivationMotivation

The runtime and performance of randomized algorithms can vary dramatically on the same instance and on different instances.

Goal: Improve the performance of different algorithms by combining them into a portfolio to exploit their relative strengths.

Carla P. Gomes


Branch & Bound:Best Bound vs. Depth First Search

Branch & Bound:Best Bound vs. Depth First Search

Carla P. Gomes


Branch & Bound(Randomized)

Branch & Bound(Randomized)

Standard OR approach for solving Mixed Integer Programs (MIPs)• Solve linear relaxation of MIP• Branch on the integer variables for which the solution of the LP relaxation is non-integer:

apply a good heuristic (e.g., max infeasibility) for variable selection ( + randomization ) and create two new nodes (floor and ceiling of the fractional value)

• Once we have found an integer solution, its objective value can be used to prune other nodes, whose relaxations have worse values

Carla P. Gomes


Branch & BoundDepth First vs. Best bound

Branch & BoundDepth First vs. Best bound

Critical in performance of Branch & Bound:

the way in which the next node to be expanded is selected.

Best-bound - select the node with the best LP bound

(standard OR approach) ---> this case is equivalent to A*, the LP relaxation provides an admissible search heuristic

Depth-first - often quickly reaches an integer solution

(may take longer to produce an overall optimal value)

Carla P. Gomes


Portfolio of AlgorithmsPortfolio of Algorithms

A portfolio of algorithm is a collection of algorithms and / or copies of the same algorithm running interleaved or on different processors.

Goal: to improve on the performance of the component algorithms in terms of:

expected computational cost“risk” (variance)

Efficient Set or Efficient Frontier: set of portfolios that are best in terms of expected value and risk.

Carla P. Gomes


Depth-First: Average - 18000;St. Dev. 30000

Brandh & Bound for MIP Depth-first vs. Best-bound

Brandh & Bound for MIP Depth-first vs. Best-boundC

um

ula

tive

Fre

qu

enci

es

Number of nodes

30%Best bound

Best-Bound: Average-1400 nodes; St. Dev.- 1300 Optimal strategy: Best Bound

45%Depth-first

Carla P. Gomes


Depth-First and Best and Bound do not dominate each other overall.

Carla P. Gomes


Heavy-tailed behavior of Depth-firstHeavy-tailed behavior of Depth-first

Carla P. Gomes


Portfolio for heavy-tailed search procedures (2 processors)

Portfolio for heavy-tailed search procedures (2 processors)

0 DF / 2 BB

2 DF / 0 BB

Standard deviation of run time of portfolios

Expect

ed r

un t

ime o

f p

ort

folio

s

Carla P. Gomes


Portfolio for 6 processorsPortfolio for 6 processors

0 DF / 6 BB

6 DF / 0BB

Exp

ecte

d r

un

tim

e of

por

tfol

ios

5 DF / 1BB

3 DF / 3 BB

4 DF / 2 BB

Efficient set


Carla P. Gomes


Portfolio for 20 processorsPortfolio for 20 processors

0 DF / 20 BB

20 DF / 0 BBExp

ecte

d r

un

tim

e of

por

tfol

ios

The optimal strategy is to run Depth First on the 20 processors!

Optimal collective behavior emerges from suboptimal individual behavior.


Carla P. Gomes


Compute Clusters and Distributed Agents

Compute Clusters and Distributed Agents

With the increasing popularity of compute clusters and distributed problem solving / agent paradigms, portfolios of algorithms --- and flexible computation in general --- are rapidly expanding research areas.

(Baptista and Marques da Silva 00, Boddy & Dean 95, Bayardo 99, Davenport 00, Hogg 00, Horvitz 96, Matsuo 00, Steinberg 00, Russell 95, Santos 99, Welman 99. Zilberstein 99)

Carla P. Gomes


Portfolio for heavy-tailed search procedures (2-20 processors)

Portfolio for heavy-tailed search procedures (2-20 processors)

Carla P. Gomes


A portfolio approach can lead to substantial improvements in the expected cost and risk of stochastic algorithms, especially in the presence of heavy-tailed phenomena.

Carla P. Gomes


Summary of RandomizationSummary of Randomization

Considered randomized backtrack search.

Showed Heavy-Tailed Distributions.

Suggests: Rapid Restart Strategy.

--- cuts very long runs

--- exploits ultra-short runs

Experimentally validated on previously unsolved planning and scheduling problems.

Portfolio of Algorithms for cases where no single heuristic dominates

Carla P. Gomes


Research Direction:Learning Restart Policies

Research Direction:Learning Restart Policies

Carla P. Gomes


Bayesian Model Structure LearningBayesian Model Structure Learning

(Horvitz, Ruan, Gomes, Kautz, Selman, Chickering 2001)

Learning to infer predictive models from data and to identify key variables==> restarts, cutoffs and other adaptive behavior of search algorithms.

Carla P. Gomes


Green - long runsGray - short runs

Variance in number of uncolored cells across rows and columns

Number uncolored cells per column

Min depth Avg Depth

Max number of uncolored cells across rows and columns

Quasigroup Order 34 (CSP)

Model accuracy 96.8% vs 48% for the marginal model

Carla P. Gomes


Analysis of different solver features and problem features

Analysis of different solver features and problem features

Carla P. Gomes


OutlineOutline


Randomization

Conclusions

Carla P. Gomes


Summary Summary

The understanding of the structural properties of problem instances based on notions such as

phase transitions, backbone, and balance provides new insights into the practical complexity of many

computational tasks.

Active research area with fruitful interactions between computer science, physics (approaches

from statistical mechanics), and mathematics (combinatorics / random structures).

Carla P. Gomes


Stochastic search methods (complete and incomplete) have been shown very effective.

Restart strategies and portfolio approaches can lead to substantial improvements in the expected runtime and variance, especially in the presence of heavy-tailed phenomena.

Randomization is therefore a tool to improve algorithmic performance and robustness.

Machine Learning techniques can be used to learn predicitive models.

Summary

Carla P. Gomes


General Solution Methods

Real WorldProblems

Exploiting Structure:Tractable Components

Transition Aware Systems(phase transitionconstrainedness

backbone resources)

RandomizationExploits variance

to improve robustness and performance

Bridging the GapBridging the Gap

Carla P. Gomes



Check also:

www.cis.cornell.edu/iisi


Check also:

www.cis.cornell.edu/iisi

Demos, papers, etc.

carla p. gomes school on optimization cpaior02 exploiting structure and randomization in...

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