Post on 22-Dec-2015
Carla P. Gomes
School on Optimization CPAIOR02
Exploiting Structure and Randomization
in Combinatorial Search
Carla P. Gomesgomes@cs.cornell.edu
www.cs.cornell.edu/gomes
Intelligent Information Systems InstituteDepartment of Computer Science
Cornell University
Exploiting Structure and Randomization
in Combinatorial Search
Carla P. Gomesgomes@cs.cornell.edu
www.cs.cornell.edu/gomes
Intelligent Information Systems InstituteDepartment of Computer Science
Cornell University
Carla P. Gomes
School on Optimization CPAIOR02
OutlineOutline
A Structured Benchmark Domain
Randomization
Conclusions
Carla P. Gomes
School on Optimization CPAIOR02
OutlineOutline
A Structured Benchmark Domain
Randomization
Conclusions
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Fiber Optic Networks
Nodesconnect point to point
fiber optic links
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Are all the Quasigroup Instances(of same size) Equally Difficult?
1820150
Time performance:
165
What is the fundamental difference between instances?
Carla P. Gomes
School on Optimization CPAIOR02
Are all the Quasigroup Instances
Equally Difficult?
1820 165
40% 50%
150
Time performance:
35%
Fraction of preassignment:
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
QCPDifferent Representations /
Encodings
QCPDifferent Representations /
Encodings
Carla P. Gomes
School on Optimization CPAIOR02
Cubic representation of QCP
Columns
Rows
Colors
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Quasigroup as SatisfiabilityQuasigroup as Satisfiability
Two different encodings for SAT:
2D encoding (or minimal encoding);
3D encoding (or full encoding);
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
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School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Structural features of instances provide insights into their hardness namely:
Backbone
Inherent Structure and Balance
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Inherent Structure and Balance
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
SATZSATZ
Balanced QCP
Rectangular QCP
Aligned QCP
QCP
QWH
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
OutlineOutline
A Structured Benchmark Domain
Randomization
Conclusions
Carla P. Gomes
School on Optimization CPAIOR02
Randomized Backtrack Search Randomized Backtrack Search ProceduresProcedures
Randomized Backtrack Search Randomized Backtrack Search ProceduresProcedures
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
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School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Randomized StrategiesRandomized Strategies
Strategy Variable sel. Value sel.
DD deterministic deterministic
DR deterministic random
RD random deterministic
RR random random
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Quasigroup DemoQuasigroup Demo
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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
School on Optimization CPAIOR02
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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School on Optimization CPAIOR02
1
Carla P. Gomes
School on Optimization CPAIOR02
75%<=30
Number backtracks Number backtracks
Pro
port
ion o
f ca
ses
Solv
ed
5%>100000
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
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School on Optimization CPAIOR02
Standard Distribution(finite mean & variance)
Power Law Decay
Exponential Decay
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Normal, Cauchy, and LevyNormal, Cauchy, and Levy
Normal - Exponential Decay
Cauchy -Power law DecayLevy -Power law Decay
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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
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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
School on Optimization CPAIOR02
The record of 10,000 tosses of an ideal coin
(Feller)
Zero crossing Long periods without zero crossing
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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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Formal Models of Heavy-Tailed Behavior in Combinatorial Search
Chen, Gomes, Selman 2001
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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;
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School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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.
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School on Optimization CPAIOR02
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)
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School on Optimization CPAIOR02
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
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Bounded Heavy-Tailed BehaviorBounded Heavy-Tailed Behavior
(show demo)
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No Heavy-tailed behavior for Proving Optimality
No Heavy-tailed behavior for Proving Optimality
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Proving OptimalityProving Optimality
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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)
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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. 97, Hoos 99, Horvitz 99, Huberman, Lukose and Hogg 97, Karp et al 96, Luby et al. 93, Rish et al. 97, Wlash 99)
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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
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RestartsRestarts
70%unsolved
1-F
(x)
Un
solv
ed f
ract
ion
Number backtracks (log)
no restarts
restart every 4 backtracks
250 (62 restarts)
0.001%unsolved
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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
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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
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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
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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)
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Retransmissions in Sequential Decoding
Retransmissions in Sequential Decoding
1-F
(x)
Un
solv
ed f
ract
ion
Number backtracks (log)
without retransmissions
with retransmissions
Gomes et al. 2000 / 20001
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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.
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Deterministic SearchDeterministic Search
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RestartsRestarts
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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
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Algorithm Portfolio DesignAlgorithm Portfolio Design
Gomes and Selman 1997 - Proc. UAI-97;
Gomes et al 1997 - Proc. CP97.
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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.
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Branch & Bound:Best Bound vs. Depth First Search
Branch & Bound:Best Bound vs. Depth First Search
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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
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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)
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School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Depth-First and Best and Bound do not dominate each other overall.
Carla P. Gomes
School on Optimization CPAIOR02
Heavy-tailed behavior of Depth-firstHeavy-tailed behavior of Depth-first
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
Standard deviation of run time of portfolios
Carla P. Gomes
School on Optimization CPAIOR02
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.
Standard deviation of run time of portfolios
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Portfolio for heavy-tailed search procedures (2-20 processors)
Portfolio for heavy-tailed search procedures (2-20 processors)
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Research Direction:Learning Restart Policies
Research Direction:Learning Restart Policies
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
Analysis of different solver features and problem features
Analysis of different solver features and problem features
Carla P. Gomes
School on Optimization CPAIOR02
OutlineOutline
A Structured Benchmark Domain
Randomization
Conclusions
Carla P. Gomes
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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
School on Optimization CPAIOR02
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.