Identify Phase Transitions In Problem Hardness Leverage Randomization In Computation

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Combinatorial Problems in Cooperative Control:

Complexity and Scalability

Carla P. Gomes and

Bart Selman

Cornell UniversityMuri MeetingJune 2002

Combinatorial Problems in Cooperative Control:

Complexity and Scalability

Carla P. Gomes and

Bart Selman

Cornell UniversityMuri MeetingJune 2002

2

Analysis of Actions & Tradeoffs

Constrai ntAnalysis

ProblemDescription

ProblemFormula tor

Constraints

ProblemSolver

ProblemSolver

Problem Reformulator

Constraints’ Structure/Critic alityMonitor / Estimator

Problem Reformulator

Phase Transitions in hardness

•Identify Phase Transitions In Problem Hardness• Leverage Randomization In Computation

Develop procedures that recognize and react to Structure in Problem Hardness

Goal

Principled dynamic control of communication and computational resources in large distributed autonomous systems, allowing for:

•Scalability•Time critical applications •Robustness guarantees

-

Overview

Overall Approach

Use findings in both the design and operation of complex (distributed)

systems

Hardness Aware Systems(Computationally)

3

OutlineOutline

ROBOFLAG Drill – Computational Issues

Capturing Structure in Combinatorial Problems

Randomization and Approximations

Conclusions

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ROBOFLAG Drill ROBOFLAG Drill

Problem is hybrid, combining discrete and continuous components, with multiple

constraints.

Overall the Roboflag control problem provides an

excellent test bed for the development of scalable

techniques for complex optimization.

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Problem RepresentationProblem Representation

ROBOFLAG Drill Formulation by Raff D’Andrea and Matt Earl.

• Represented as a mixed logical system (MLD) in which the objective is to compute optimal control policies that minimize the total score of the game.

Mathematical Formulation of the Optimization Problem Mixed Integer Linear Program

6

.

We are investigating how to scale up solutionsof the ROBOFLAG Drill focusing on:

- Mixed Integer Program (MIP) formulations- Randomization and Approximation methods- Combining MIP and constraint search

techniques.- Portfolios of Algorithms

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Scaling Up Mixed Integer Linear Program Formulations (MILP)

Scaling Up Mixed Integer Linear Program Formulations (MILP)

Standard approach for solving MILP:

Branch and Bound

How can we improve upon Branch and Bound strategies?

Ideas:

Different search strategies for node selection

Randomization

Portfolios of algorithms

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Branch & Bound:Depth First vs. Best bound

Branch & Bound:Depth First vs. Best bound

Critical to performance of Branch & Bound is the way

in which the next node to be expanded is selected.

Standard approach:

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

Alternative:

Depth-first --- often quickly reaches an integer solution

(may take longer to produce an overall optimal value)

Tradeoffs between these choices depend on underlying

problem stucture (Gomes et al. 2001).

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ROBOFLAG TestbedROBOFLAG Testbed

Hybrid node selection - Best Bound and Depth First

Depth First search works well.

Problems that could not be solved before with best bound using were solved with depth first.

Current largest problem solved with CPLEX using Depth First Search (8 attackers and 3 defenders):

• Integer variables = 4040

• Continuous variables 400

• Constraints - 13580 constraints

• Time - 244 secs

(Matt Earl 2002)

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Much room for improvement…Much room for improvement…

We are not yet using other problem formulations,

Nor are we yet exploiting randomization and parallelism.

Doing so should allow us to solve problems at

least one or two orders of magnitude larger.

(100,000 to 500,000 vars and 1,000,000+

constraints)

Also, we should be able to include more complex constraints.

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the importance of problem representation…


the importance of problem representation…

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Completing Latin Squares:An Abstraction for Real World Applications

Completing Latin Squares:An Abstraction for Real World Applications

Latin Square(Order 4)

32% preassignment

Gomes and Selman 96

A Latin Square is an n-by-n matrix such that each row and column is a permutation of the same n colors

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Switches in Fiber Optic NetworksSwitches in Fiber Optic Networks

Dynamic wavelength routing in Fiber Optic Networks can be directly mapped into the Latin Square 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

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LP Based Formulations

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Assignment FormulationAssignment Formulation

Cubic representation of QCP

Columns

Rows

Colors

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QCPAssignment Formulation

QCPAssignment Formulation

}1,0{ijkx

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

nkjii ijkx

kj

....,,2,1,,1,, nkjik ijkxji

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

nkjij ijkx

ki

Row/color line

Column/color line

Row/column line

kijPLStskjikji

x ..,,

1,,

nijkxnn

j ki 1 11

max

....,,2,1,,;, nkjikcolorhasjicellijkx

Max number of colored cells

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Using a MIP formulation and Branch and Bound we can only find solutions for Latin Squares up to Order 15 (15 x 15)

we can do better, even with an LP based formulation using a less obvious encoding

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Packing formulationPacking formulation

Max number of colored cells in the selected patterns

s.t. one pattern per family

a cell is covered at most by one pattern

Families of patterns

(partial patterns are not shown)

Gomes and Shmoys 2002

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QCPPacking Formulation

QCPPacking Formulation

MkMky

,},1,0{

,

1,

kMM Mk

yk one pattern per color

at most one pattern covering each cell

kMM MkyMn

k ,||1

max

1),:( ,1

,

Mji

kMM Mk

yn

kji

Max number of colored cells

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Any feasible solution to the packing LP relaxation is

also a solution to the assignment LP relaxation

The value of the assignment relaxation is at least the bound implied by the packing formulation => the packing formulation provides a tighter upper bound than the assignment formulation

Limitation – size of formulation is exponential in n. (one may apply column generation techniques)

k

MM Mky

ijkx

,

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ApproximationBased on Packing Formulation

ApproximationBased on Packing Formulation

Randomization scheme:

for each color K choose a pattern with probability (so that some matching is selected for each color)

As a result we have a pattern per color.

Problem: some patterns may overlap, even though in expectation, the constraints imply that the number of matchings in which a cell is involved is 1.

*,Mk

y

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Packing formulationPacking formulation

0.8

0.2

1

1 1

Max number of colored cells in the selected patterns

s.t. one pattern per family

a cell is covered at most by one pattern

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(1-1/e)- ApproximationBased on Packing Formulation


Let’s assume that the PLS is completable

Z*=h

What is the expected number of cells uncolored by our randomized procedure due to overlapping conflicts?

From we can compute

So, the desired probability corresponds to the probability of a cell not be colored with any color, i.e.:

*,Mk

y

kMM

Mky

ijkx

*

,*

)*1()*2

1)(*1

1()1

*1( ijnxijx

ijx

n

k ijkx

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This expression is maximized when all the

are equal therefore:

So the expected number of uncolored cells is at most at least holes are expected to be filled by this technique.

*ijkx

)*1()*2

1)(*1

1()1

*1( ijnxijx

ijx

n

k ijkx

enn

n

k ijkx 1)11()

1*1(

eh he)11(

1- 1/e ~ 0.632 - This is a very good guaranteefor a polynomial time algorithm!

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Another Formulation

Constraint Satisfaction Problem

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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

( njjnxjx

jxalldiff

)2(nO

)(nO

row

column

kijPLStsjikjix ..,,

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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

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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

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Pure CSP approaches solve QCP instances up

to order 33 (1089 variables) relatively well.

(LP based – only up to order 15 – 125 variables)

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We are exploring more direct encodings for the ROBOFLAG DRILL

Representations avoiding discretization based on time.

constraint based abstractions closer to the physical system, e.g., based movements / trajectories.

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Randomization

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BackgroundBackground

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

Simulated annealing

Genetic algorithms

Tabu Search

Walksat and variants.

Limitation: inherent incomplete nature of local search methods.

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Randomized variable and/or value selection – lots of different ways.

Example: randomly breaking ties in variable and/or value selection.

Compare with standard lexicographic tie-breaking.

Note: No problem maintaining the completeness of the algorithm!

Randomized backtrack searchRandomized backtrack search

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Sa

mp

le m

ea

nErratic Behavior of Mean

Number runs

Empirical Evidence of Heavy-TailsEmpirical Evidence of Heavy-Tails

(*) no solution found - reached cutoff: 2000Time: (*)3011 (*)7

Easy instance – 15 % preassigned cells

Gomes et al. 97

500

2000

3500

Median = 1!

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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

Infinite variance, infinite mean

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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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Using randomization and restarts we can solve considerably larger instances up to order QCP instances up to order 40 (1600 variables).

Note: this problem is highly exponential – instances of order 40 are much more difficult than instances of order 33!

We are also experimenting with randomization in the ROBOFLAG DRILL

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Hybrid MIP/CSP Approaches

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CSP Model

LP Model + LP Randomized Rounding

Heavy-tails

We want to maintain completeness

How do we combine all these ingredients?

A HYBRID COMPLETE CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH

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HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH


Central features of algorithm:

• Complete Backtrack search algorithm

• It maintains two formulations

• CSP model

• Relaxed LP model

• LP Randomized rounding for setting values at the top of the tree

• CSP + LP inference

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Variable setting

controlled by LP Randomized

RoundingCSP & LP Inference

Search & Inferencecontrolled by CSP

%LP

Interleave-LP



•Populate CSP Model• Perform propagation

•Populate LP solver•Solve LP

Adaptive CUTOFF

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Empirical Results

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Time PerformanceTime Performance

Order 35

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PerformancePerformance

With the hybrid strategy we also solve instances of order 40 in critically constrained area – out of reach for pure CSP;

We even solved a few balanced instances of order 50 in the critically constrained order!

• more systematic experimentation is required to better understand limitations and strengths of approach.

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Conclusions

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ConclusionsConclusions

Approximations based on LP randomized rounding (variable/value setting) + constraint propagation --- very powerful.

Combatting heavy-tails of backtrack search through randomization.

Consequence: New ways of designing algorithms ---

aim for strategies which have highly asymmetric distributions that can be exploited using restarts, portfolios of algorithms, and interleaved/parallel runs.

General approach --- holds promise for a range of hard combinatorial problems.

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Scaling up ROBOFLAG - Other Formulations for Solving the Control Optimization Problem

Scaling up ROBOFLAG - Other Formulations for Solving the Control Optimization Problem

Encodings that provide “tighter” relaxations for the LP problem.

Approximate representations using abstractions (“synthesize larger movements / trajectories”). Avoid discretization based on time. Less compact representations may allow for more propagation and scale up better.

Constraint Satisfaction Problem (CSP) formulations.Hybrid CSP/LP formulations.Approximations based on LP randomized rounding.

Goal: At least two orders of magnitude scale-up over current state-of-the-art.

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www.cs.cornell.edu/gomes

Check also:

www.cis.cornell.edu/iisi

www.cs.cornell.edu/gomes

Check also:

www.cis.cornell.edu/iisi

Demos, papers, etc.

Documents

Identify Phase Transitions In Problem Hardness Leverage Randomization In Computation