Exploiting Structure and Randomization in Combinatorial Search Carla P. Gomes [e-mail] www.cs.cornell.edu/gomes Intelligent Information Systems Institute Department of Computer Science Cornell University

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Exploiting Structure and Randomization in
Combinatorial SearchCarla P. Gomesgomes_at_cs.corn
ell.eduwww.cs.cornell.edu/gomesIntelligent
Information Systems InstituteDepartment of
Computer ScienceCornell University

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Outline

• A Structured Benchmark Domain
• Randomization
• Conclusions

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Outline

• A Structured Benchmark Domain
• Randomization
• Conclusions

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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)
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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)
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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 )
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Fiber Optic Networks
Nodes connect point to point fiber optic links
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Fiber Optic Networks
Nodes connect point to point fiber optic links
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Routing in Fiber Optic Networks
Input Ports
Output Ports
1
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.
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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)
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Traditional View of Hard Problems - Worst Case
View

• Theyre NP-Completetheres no way to do
  anything but try heuristic approaches and hope
  for the best.

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New Concepts in Computation

• Not all NP-Hard problems are the same!
• We now have means for discriminating easy from
  hard instances
• ---gt Phase Transition concepts

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NP-completeness is a worst-case 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?
What is the fundamental difference between
instances?
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Are all the Quasigroup Instances Equally
Difficult?
1820
165
50
40
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Complexity of Quasigroup Completion
Median Runtime (log scale)
Fraction of pre-assignment
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Phase Transition
Fraction of unsolvable cases
Fraction of pre-assignment
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• 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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QCPDifferent Representations / Encodings
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Rows
Colors
Columns
Cubic representation of QCP
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QCP as a MIP

• Variables -
• Constraints -

Row/color line
Column/color line
Row/column line
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QCP as a CSP

• Variables -
• Constraints -

vs. for MIP
row
column
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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 )
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Exploiting Structure in QCP ALLDIFF as Global
Constraint
(Berge 70, Regin 94, Shaw and Walsh 98 )
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Exploiting Structure Arc Consistency vs. All Diff
AllDiff Solves up to order 33 Size search space

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Quasigroup as Satisfiability

• Two different encodings for SAT
• 2D encoding (or minimal encoding)
• 3D encoding (or full encoding)

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2D 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)

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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.
• 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

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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

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SATZ on 2D encoding (Order 20 -28)
Order 28
1,000,000
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
backtrack and much higher orders can be solved
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Walksat on 2D and 3D encoding(Order 30-33)
1,000,000
2D order 33
3D 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
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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

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• 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
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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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New Phase Transition in Backbone QCP (satisfiable
instances only)
Backbone
of Backbone
Computational cost
Fraction of preassigned cells
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• Inherent Structure and Balance

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Quasigroup Patterns and Problems Hardness
Tractable
Very hard
(Kautz, Ruan, Achlioptas, Gomes, Selman 2001)
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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, SATZ, and SATO Balacing makes
instances harder
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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).

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Outline

• A Structured Benchmark Domain
• Randomization
• Conclusions

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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.

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Background

• We want to explore the addition of a
• stochastic element to a systematic search
• procedure without losing completeness.

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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.

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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.

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Randomized Strategies

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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 CostQuasigroup
Completion Problem
3500!
sample mean
Median 1!
number of runs
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1
53
75lt30
5gt100000
Proportion of cases Solved
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Heavy-Tailed Distributions

• infinite variance infinite mean
• Introduced by Pareto in the 1920s
• --- probabilistic curiosity.
• Mandelbrot established the use of heavy-tailed
  distributions to model real-world fractal
  phenomena.
• Examples stock-market, earth-quakes, weather,...

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Decay of Distributions

• Standard --- Exponential Decay
• e.g. Normal
• Heavy-Tailed --- Power Law Decay
• e.g. Pareto-Levy

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Normal, Cauchy, and Levy
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Tail Probabilities (Standard Normal, Cauchy,
Levy)

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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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Heavy-tails vs. Non-Heavy-Tails
Normal (2,1000000)
1-F(x) Unsolved fraction
O,1gt200000
Normal (2,1)
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
• infinite mean and
  infinite variance
• infinite variance

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Heavy-Tailed Behavior in QCP Domain
(1-F(x))(log) Unsolved fraction
Number backtracks (log)
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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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• T(n) - the number of leaf nodes visited
• - choice at level i (1 - bad choice 0
  -good choice)
• (note there is exactly one choice of zero-one
  assignments to the variables for each
  possible value of T(n) any such assignment has
  probability .
• T(n) follows an Uniform distribution

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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
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Balanced 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.

Chen, Gomes Selman 01
70

• 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.
• ---gt resulting trees may vary dramatically from
  run to run.

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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

(show demo)
b 2
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• Expected Run Time
• (infinite expected time)
• Variance
• (infinite variance)
• Tail
• (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
Interleaved (1 machine) 10 x 1 10 seconds
5 x speedup
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Restarts
70 unsolved
no restarts
1-F(x) Unsolved fraction
restart every 4 backtracks
0.001 unsolved
250 (62 restarts)
Number backtracks (log)
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Example of Rapid Restart Speedup(planning)
Number backtracks (log)
Cutoff (log)
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Sketch of proof of elimination of heavy tails

• Lets truncate the search procedure
• after m backtracks.
• Probability of solving problem with truncated
  version
• Run the truncated procedure and restart it
  repeatedly.

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Y - does not have Heavy Tails
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Decoding in Communication Systems
84
Retransmissions in Sequential Decoding
without retransmissions
1-F(x) Unsolved fraction
with retransmissions
Number backtracks (log)
Gomes et al. 2000 / 20001
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Paramedic 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 Search
87
Restarts
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Results on Effectiveness of Restarts
Deterministic

() not found after 2 days
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Algorithm Portfolio Design

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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.

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Branch BoundBest Bound vs. Depth First
Search
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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

• 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) ---gt
• 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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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.

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Brandh Bound for MIP Depth-first vs.
Best-bound
Depth-First Average - 18000St. Dev. 30000
96

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

97
Heavy-tailed behavior of Depth-first
98
Portfolio for heavy-tailed search procedures (2
processors)
2 DF / 0 BB
Expected run time of portfolios
0 DF / 2 BB
Standard deviation of run time of portfolios
99
Portfolio for 6 processors
0 DF / 6 BB
Expected run time of portfolios
6 DF / 0BB
Standard deviation of run time of portfolios
100
Portfolio for 20 processors
0 DF / 20 BB
The optimal strategy is to run Depth First on
the 20 processors!
Expected run time of portfolios
Optimal collective behavior emerges from
suboptimal individual behavior.
20 DF / 0 BB
Standard deviation of run time of portfolios
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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)
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Portfolio for heavy-tailed search procedures
(2-20 processors)
103

• 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.

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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

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Research DirectionLearning Restart Policies

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Bayesian Model Structure Learning
Learning to infer predictive models from data and
to identify key variables gt restarts, cutoffs
and other adaptive behavior of search algorithms.
(Horvitz, Ruan, Gomes, Kautz, Selman, Chickering
2001)
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Quasigroup Order 34 (CSP)
Min depth
Avg Depth
Variance in number of uncolored cells across
rows and columns
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
108
Analysis of different solver features and problem
features
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Outline

• A Structured Benchmark Domain
• Randomization
• Conclusions

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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).

111

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.

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Bridging the Gap
Exploiting Structure Tractable
Components Transition Aware Systems (phase
transition constrainedness backbone
resources) Randomization Exploits variance to
improve robustness and performance
General Solution Methods
Real World Problems
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