Exploiting Structure and
Randomization
in Combinatorial Search
Carla P. Gomes
[email protected]
www.cs.cornell.edu/gomes
Intelligent Information Systems Institute
Department of Computer Science
Cornell University
Carla P. Gomes
School on Optimization
CPAIOR02
Outline
A Structured Benchmark Domain
Randomization
Conclusions
Carla P. Gomes
School on Optimization
CPAIOR02
Outline
A Structured Benchmark Domain
Randomization
Conclusions
Carla P. Gomes
School on Optimization
CPAIOR02
Quasigroups or Latin Squares:
An Abstraction for Real World Applications
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)
Carla P. Gomes
School on Optimization
CPAIOR02
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 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
Nodes
connect point to point
fiber optic links
Carla P. Gomes
School on Optimization
CPAIOR02
Fiber Optic Networks
Nodes
connect point to point
fiber optic links
Each fiber optic link supports a
large 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
preassigned channels
Input Ports
1
Output Ports
1
2
2
3
3
4
4
Routing Node
How can we achieve conflict-free routing in each node of the network?
Dynamic wavelength routing is a NP-hard problem.
Carla P. Gomes
School on Optimization
CPAIOR02
QCP Example Use: Routers in
Fiber Optic Networks
Dynamic wavelength routing in Fiber Optic Networks can be
directly mapped into the Quasigroup Completion Problem.
•each channel cannot be repeated in the same input port
(row constraints);
• each channel cannot be repeated in the same output
port (column constraints);
1
2
3
4
Output ports
Output Port
1
2
3
4
Input ports
Input Port
CONFLICT FREE
LATIN ROUTER
(Barry and Humblet 93, Cheung et al. 90, Green 92, Kumar et al. 99)
Carla P. Gomes
School on Optimization
CPAIOR02
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 Computation
Not all NP-Hard problems are the
same!
We now have means for
discriminating easy from hard
instances
---> Phase Transition concepts
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School on Optimization
CPAIOR02
NP-completeness is a worstcase notion – what about average
complexity?
Structural differences
between instances of the same
NP- complete problem (QCP)
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Are all the Quasigroup Instances
(of same size) Equally Difficult?
Time performance:
150
1820
165
What is the fundamental difference between instances?
Carla P. Gomes
School on Optimization
CPAIOR02
Are all the Quasigroup Instances
Equally Difficult?
Time performance:
150
Fraction of preassignment:
1820
165
35%
40%
Carla P. Gomes
50%
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CPAIOR02
Median Runtime (log scale)
Complexity of Quasigroup
Completion
Critically constrained area
Underconstrained
area
Overconstrained area
20%
42%
50%
Fraction
of pre-assignment
Carla P. Gomes
School on Optimization
CPAIOR02
Complexity
Graph
Phase
Transition
Fraction of unsolvable cases
Phase transition
from almost all solvable
to almost all unsolvable
Almost all solvable
area
Almost all unsolvable
area
Fraction of pre-assignment
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)
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School on Optimization
CPAIOR02
QCP
Different Representations /
Encodings
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School on Optimization
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Rows
Colors
Columns
Cubic representation of QCP
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School on Optimization
CPAIOR02
QCP as a MIP
O(n3)
cell i, j has color k; i, j,k 1, 2, ...,n.
• Variables -
x
ijk
•
x  {0,1}
ijk
Constraints - O(n2)
Row/color line

x  1 i, j,k 1, 2, ...,n.

j,k
ijk
i
Column/color line
  x  1 i, j,k 1, 2, ...,n.
i,k
ijk
j
Row/column line
 ,  x  1 i, j,k 1, 2, ...,n.
i, j
ijk
k
Carla P. Gomes
School on Optimization
CPAIOR02
QCP as a CSP
• Variables -
•
O(n2) [ vs. O(n3) for MIP]
x color of cell i, j; i, j 1, 2, ...,n.
i, j
x  {1, 2, ...,n}
i, j
Constraints - O(n) [ vs. O(n2) for MIP]
alldiff (x , x ,..., x ); i 1, 2, ...,n.
i,n
i,1 i,2
row
alldiff (x , x ,..., x ); j 1, 2, ...,n. column
n, j
1, j 2, j
Carla P. Gomes
School on Optimization
CPAIOR02
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,
Carla P. Gomes
Ottososon & Thorsteinsson 00, Refalo 99, Regin 94 )
School on Optimization
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Exploiting Structure in QCP
ALLDIFF as Global Constraint
Matching on
a Bipartite graph
Two solutions:
All-different constraint
we can update the
domains of the column
variables
(Berge 70, Regin 94, Shaw and Walsh 98 )
Analogously, we can
update the domains
of the other variables
Carla P. Gomes
School on Optimization
CPAIOR02
Exploiting Structure
Arc Consistency vs. All Diff
Arc Consistency
AllDiff
Solves up to order 20
Size search
space 20400
Solves up to order 33
Size search
space 331089
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School on Optimization
CPAIOR02
Quasigroup as Satisfiability
Two different encodings for SAT:
2D encoding (or minimal encoding);
3D encoding (or full encoding);
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School on Optimization
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2D Encoding or Minimal Encoding
3
Variables: n
x cell i, j has color k; i, j,k 1, 2, ...,n.
ijk
x  {0,1}
ijk
Each variables represents a color assigned to a cell.
Clauses:
O(n4)
• Some color must be assigned to each cell (clause of length n);
 (x  x x )
ij ij1 ij2 ijn
• No color is repeated in the same row (sets of negative binary clauses);
 (x  x )  (x  x ) (x  x )
ik i1k
i2k
i1k
i3k
i1k
ink
• No color is repeated in the same column (sets of negative binary clauses);

(x  x )  (x  x ) (x  x )
jk 1 jk
2 jk
1 jk
3 jk
1 jk
njk
Carla P. Gomes
School on Optimization
CPAIOR02
3D 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.
O(n4)
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;
Carla P. Gomes
School on Optimization
CPAIOR02
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)
1,000,000
Order 28
Order 20
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
Carla P. Gomes
backtrack and much higher orders can be solved;
School on Optimization
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Walksat on 2D and 3D encoding
(Order 30-33)
1,000,000
3D order 33
2D 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 - 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
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Backbone
Backbone is the shared structure of all the
solutions to a given instance.
This instance has
4 solutions:
Backbone
Total number of backbone variables: 2
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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)
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School on Optimization
CPAIOR02
New Phase Transition in Backbone
QCP (satisfiable instances only)
% of Backbone
% Backbone
Sudden phase transition in Backbone
Computational
cost
Fraction of preassigned cells
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School on Optimization
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Inherent Structure and Balance
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Quasigroup Patterns and
Problems Hardness
Rectangular Pattern
Aligned Pattern
Tractable
(Kautz, Ruan, Achlioptas, Gomes, Selman 2001)
Balanced Pattern
Very hard
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School on Optimization
CPAIOR02
SATZ
Balanced QCP
Rectangular QCP
QCP
QWH
Aligned QCP
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Walksat
Balanced filtered QCP
Balance QWH
QCP
QWH
aligned
rectangular
We observe the same ordering in hardness when using Walksat,
Carla P. Gomes
SATZ, and SATO – Balacing makes instances harder
School on Optimization
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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
Outline
A Structured Benchmark Domain
Randomization
Conclusions
Carla P. Gomes
School on Optimization
CPAIOR02
Randomized Backtrack Search
Procedures
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Background
Stochastic strategies have been very successful
in the area of local search.
Simulated annealing
Genetic algorithms
Tabu Search
Gsat and variants.
Limitation: inherent incomplete nature of local
search methods.
Carla P. Gomes
School on Optimization
CPAIOR02
Background
We want to explore the addition of a
stochastic element to a systematic search
procedure without losing completeness.
Carla P. Gomes
School on Optimization
CPAIOR02
Randomization
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.
Carla P. Gomes
School on Optimization
CPAIOR02
Randomization
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 Strategies
Strategy
Variable sel.
Value sel.
DD
deterministic
deterministic
DR
deterministic
random
RD
random
deterministic
RR
random
random
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CPAIOR02
Quasigroup Demo
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Distributions of Randomized
Backtrack Search
Key Properties:
I Erratic behavior of mean
II Distributions have “heavy tails”.
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Erratic Behavior of Search Cost
Quasigroup Completion Problem
3500!
sample
mean
2000
Median = 1!
500
number of runs
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CPAIOR02
1
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Proportion of cases Solved
75%<=30
Number backtracks
5%>100000
Number backtracks
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Heavy-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,...
Carla P. Gomes
School on Optimization
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Decay of Distributions
Standard --- Exponential Decay
e.g. Normal:
Pr[ X  x] Ce  x2, for some C  0, x 1
Heavy-Tailed --- Power Law Decay
e.g. Pareto-Levy:
Pr[ X  x] Cx  , x  0
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Power Law Decay
Exponential Decay
Standard Distribution
(finite mean & variance)
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Normal, Cauchy, and Levy
Cauchy -Power law Decay
Levy -Power law Decay
Normal - Exponential Decay
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Tail Probabilities
(Standard Normal, Cauchy, Levy)
c
Normal
0
0.5
1
0.1587
2
0.0228
3
0.001347
4 0.00003167
Cauchy Levy
0.5
1
0.25
0.6827
0.1476
0.5205
0.1024
0.4363
0.078
0.3829
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School on Optimization
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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.
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Zero crossing
Long periods without
zero crossing
The record of 10,000 tosses of an ideal coin
(Feller)
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Heavy-tails vs. Non-Heavy-Tails
1-F(x)
Unsolved fraction
50%
Random Walk
Median=2
Normal
(2,1000000)
O,1%>200000
Normal
(2,1)
2
X - number of steps the walk takes to return to zero (log scale)
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How to Check for “Heavy Tails”?
Log-Log plot of tail of distribution
should be approximately linear.
Slope gives value of

 1
infinite mean and infinite variance
1  2
infinite variance
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(1-F(x))(log)
Unsolved fraction
Heavy-Tailed Behavior in QCP Domain
  0.153
  0.319
18%
unsolved
  0.466
 1 => Infinite mean
Number backtracks (log)
0.002%
unsolved
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School on Optimization
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Formal Models of Heavy-Tailed Behavior in
Combinatorial Search
Chen, Gomes, Selman 2001
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Motivation
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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Balanced vs. Imbalanced
Tree Model
Balanced Tree Model:
• chronological backtrack search model;
• fixed variable ordering;
• random child selection with no propagation
mechanisms;
(show demo)
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School on Optimization
CPAIOR02
n
1

2
E[T (n)]
2

























2n 1
2
V [T (n)]
12









The run time distribution of chronological backtrack search on
a complete balanced tree is uniform (therefore not heavy-tailed).
Both the expected run time and variance scale exponentially
Carla P. Gomes
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Balanced Tree Model
n
1

2
E[T (n)]
2
















V [T (n)]









2
n
2 1
12









• 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.
Chen, Gomes & Selman 01
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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
School on Optimization
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Formal Model Yielding
Heavy-Tailed Behavior
T - the number of leaf nodes visited up to and
including the successful node; b - branching
factor
P[T bi ] (1 p) pi i  0
(show demo)
b=2
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Expected Run Time
p  1 E[T ]
b
Variance
Tail
p  1 V [T ]
b2
p 1
2
b
(infinite expected time)
(infinite variance)
log p
P[T  L ] p2 L b C L   2
(heavy-tailed)
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Bounded Heavy-Tailed Behavior
(show demo)
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No Heavy-tailed behavior for Proving
Optimality
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Proving Optimality
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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 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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Super-linear Speedups
X
10
X
10
X
10
X
10
X
10
solved
Sequential: 50 +1 = 51 seconds
Parallel: 10 machines --- 1 second
51 x speedup
Interleaved (1 machine): 10 x 1 = 10 seconds
5 x speedup
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Restarts
1-F(x)
Unsolved fraction
no restarts
70%
unsolved
restart every 4 backtracks
0.001%
unsolved
250 (62 restarts)
Number backtracks (log)
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Example of Rapid Restart Speedup
(planning)
100000
log ( backtracks )
Number backtracks (log)
1000000
100000
~10 restarts
10000
~100 restarts
2000
1000
1
20
10
100
1000
10000
100000
1000000
log( cutoff )
Cutoff (log)
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Sketch of proof of elimination of
heavy tails
X  numberof backtracks to solve the problem
Let’s truncate the search procedure
after m backtracks.
Probability of solving problem with truncated version:
pm  Pr[ X  m]
Run the truncated procedure and restart it repeatedly.
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Y  total number backtracks with restarts
Number of Re starts  Y / m ~ Geometric( pm)




F  Pr[Y  y]  (1 pm)












Y /m
 c1ec2 y
Y - does not have Heavy Tails
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Decoding in Communication
Systems
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
Source
Encoder
Processing prior to transmission,
e.g., insertion of redundancy to
combat the channel noise.
Channel
Decoder
Destination
Processing of the channel output with the
objective of producing at the destination
an 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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CPAIOR02
Retransmissions in Sequential
Decoding
1-F(x)
Unsolved fraction
without retransmissions
with retransmissions
Number backtracks (log)
Gomes et al. 2000 / 20001
Carla P. Gomes
School on Optimization
CPAIOR02
Paramedic Crew Assignment
Paramedic crew assignment is the problem of assigning paramedic crews
Carla P. Gomes
from different stations to cover a given region, given several resource constraints.
School on Optimization
CPAIOR02
Deterministic Search
Carla P. Gomes
School on Optimization
CPAIOR02
Restarts
Carla P. Gomes
School on Optimization
CPAIOR02
Results on Effectiveness of Restarts
Deterministic
Logistics Planning
Scheduling 14
Scheduling 16
Scheduling 18
Circuit Synthesis 1
Circuit Synthesis 2
108 mins.
411 sec
---(*)
---(*)
---(*)
---(*)
3
R
95 sec.
250 sec
1.4 hours
~18 hrs
165sec.
17min.
(*) not found after 2 days
Carla P. Gomes
School on Optimization
CPAIOR02
Algorithm Portfolio Design
Gomes and Selman 1997 - Proc. UAI-97;
Gomes et al 1997 - Proc. CP97.
Carla P. Gomes
School on Optimization
CPAIOR02
Motivation
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
School on Optimization
CPAIOR02
Branch & Bound:
Best Bound vs. Depth First Search
Carla P. Gomes
School on Optimization
CPAIOR02
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
School on Optimization
CPAIOR02
Branch & Bound
Depth 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
School on Optimization
CPAIOR02
Portfolio 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
Cumulative Frequencies
Brandh & Bound for MIP
Depth-first vs. Best-bound
Optimal strategy: Best Bound
Best-Bound: Average-1400 nodes; St. Dev.- 1300
Depth-first
45%
30%
Best bound
Depth-First: Average - 18000;St. Dev. 30000
Number of nodes
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-first
Carla P. Gomes
School on Optimization
CPAIOR02
Expected run time of portfolios
Portfolio for heavy-tailed search
procedures (2 processors)
2 DF / 0 BB
0 DF / 2 BB
Standard deviation of run time of portfolios
Carla P. Gomes
School on Optimization
CPAIOR02
Expected run time of portfolios
Portfolio for 6 processors
0 DF / 6 BB
3 DF / 3 BB Efficient set
4 DF / 2 BB
6 DF / 0BB
5 DF / 1BB
Standard deviation of run time of portfolios
Carla P. Gomes
School on Optimization
CPAIOR02
Expected run time of portfolios
Portfolio for 20 processors
0 DF / 20 BB
The optimal strategy is to run
Depth First on the 20 processors!
Optimal collective behavior emerges
from suboptimal individual behavior.
20 DF / 0 BB
Standard deviation of run time of portfolios
Carla P. Gomes
School on Optimization
CPAIOR02
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)
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 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
Carla P. Gomes
School on Optimization
CPAIOR02
Bayesian Model Structure Learning
Learning to infer predictive models from data and to identify key variables
==> restarts, cutoffs and other adaptive behavior of search algorithms.
(Horvitz, Ruan, Gomes, Kautz, Selman, Chickering 2001)
Carla P. Gomes
School on Optimization
CPAIOR02
Quasigroup Order 34 (CSP)
Variance in number of uncolored
cells across rows and columns
Min depth
Avg Depth
Number uncolored
cells per column
Max number of uncolored
cells across rows and columns
Green - long runs
Gray - short runs
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
Carla P. Gomes
School on Optimization
CPAIOR02
Outline
A Structured Benchmark Domain
Randomization
Conclusions
Carla P. Gomes
School on Optimization
CPAIOR02
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
Summary
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.
Carla P. Gomes
School on Optimization
CPAIOR02
Bridging the Gap
General Solution
Methods
Exploiting Structure:
Tractable Components
Transition Aware Systems
(phase transition
constrainedness
backbone resources)
Randomization
Exploits variance
to improve robustness
and performance
Real World
Problems
Carla P. Gomes
School on Optimization
CPAIOR02
Demos, papers, etc.
www.cs.cornell.edu/gomes
Check also:
www.cis.cornell.edu/iisi
Carla P. Gomes
School on Optimization
CPAIOR02