Alan M. Frisch

1
Constraint Technology for the Masses

• Alan M. Frisch
• Artificial Intelligence Group
• Department of Computer Science
• University of York
• Collaborators
• Ian Miguel, Toby Walsh, Pierre Flener,
  Brahim Hnich, Zeynep Kiziltan, Justin Pearson

2
Constraint Programming is Important

• Useful for solving a wide range of problems
  including scheduling, allocation, layout.
• Often used on large, complex problems whose
  solution is crucial.
• Constraint technology is a big business.
• Powerful toolkits are available
• Eclipse from IC PARC
• Solver from ILOG

3
Formulation Bottleneck

• Use of current technology requires modelling a
  problem as a constraint program.
• This requires moderate/great expertise.
• Hence constraint technology is used almost
  exclusively by large enterprises on problems with
  large payoff.

4
Reducing the Formulation Bottleneck

• Systematise the knowledge of the expert.
• The expert can recognise patterns in problems and
  match them to patterns of problem formulations.
• The expert can choose among alternatives.

5
Exploiting Common Patterns

• Published in journals with analyses of
  properties.
• Catalogued in textbooks with guidance on
  selection.
• Implemented in libraries.
• Supported by high-level languages.
• Exploited by intelligent compilers.

This would bring constraint technology to BScs.
6
Making Constraint Technology Invisible

• Conceal constraint technology behind intuitive
  user interfaces.
• Pre-programmed patterns would be provided.

This would bring constraint technology
to the masses.
7
The Constraint Satisfaction Problem

• An instance of the CSP consists of
• Finite set of variables X1,,Xn, having finite
  domains D1,,Dn.
• Finite set of constraints. Each restricts the
  values that the variables can simultaneously
  take. Example x neq y. xyltz.

8
Solutions of a CSP Instance

• A total instantiation maps each variable to an
  element in its domain.
• A solution to a CSP instance is a total
  instantiation that satisfies all the constraints.
• Problem Given an instance
• Determine if it is satisfiable (has a solution)
• Find a solution
• Find all solutions
• Find optimal solution

9
Partial Instantiation Search

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Matrix Models

• A problem formulation involving one or more
  (multi-dimensional) matrices of decision
  variables.
• Often has row and column symmetry.

Round Robin Tournament Schedule
11
Matrix Models are Prevalent

• Commonly used to implement sets, subsets,
  partitions, relations, functions, permutations
• Of 31 problems in CSPlib (www.csplib.org), 27
  have natural matrix models, most already
  published and arguably the most natural models.
• Many have row and column symmetries.

12
Overview

• I Systematic refinement of high-level
• specifications to matrix models.
• II Breaking row and column symmetry.
• III Vision of constraint spreadsheets.

13
Part ISystematic refinement of high-level
specificationsto matrix models.
14
Balanced Academic Curriculum Problem

• Assign courses to periods in which they will be
  taught such that
• Course load in each period is not too large or
  too small.
• Credit load in each period is not too large or
  too small.
• The prerequisites to each course are taught
  before the course.

15
High-level Specification

• Find a function CUR courses ? periods
• Constraints are inequalities involving CUR,
  CUR-1, cardinality and summation.
• We have semi-systematically transformed this to a
  variety of alternative matrix models. Major
  decisions
• How to represent CUR.
• The introduction of a 1D matrix of variables to
  store the credit load of each period (from
  CGRASS).

16
1D Matrix Representation of a Function
CUR1
Periods
Periods
Periods
Periods
Periods
Periods
No straightforward way to express the credit
load of a period with facilities of current
constraint programming languages.
Courses
17
2D Matrix Representation of a Function
?1 ?1
All constraints can be expressed as linear
inequalities.
18
1D 2D Matrix Representation of a Function
CUR1
Periods
Periods
Periods
Periods
Periods
Periods
All constraints can be expressed
straightforwardly.
CUR2c,p 1 ? CUR1c p
19
Golumb Ruler Problem

• We have systematically transformed a high-level
  specification of the problem into an efficient
  CSP formulation (roughly that of Smith, Stergiou
  Walsh). This involves
• Introducing new variables.
• Replacing constraints with more powerful ones.
• Refinement choosing subset to choosing values.
• Breaking symmetry.
• Reasoning about ordered subscripts.

20
CGRASS

• A system that automatically generates implied
  constraints and symmetry breaking constraints to
  improve efficiency of solver.
• Currently operates on problem instances, but is
  now being extended to handle problem classes.
• Major lesson inferences must be made at the
  appropriate level of abstraction. Refinement
  rules as well as inference rules are needed.

21
Part IIBreaking Row and Column Symmetries
22
Index Symmetry in One Dimension
A B C
D E F
G H I

• Indistinguishable Rows

• 2 Dimensions
• A B C ? lex D E F ? lex G H I

• Can be extended to n dimensions and partial
  symmetry.

23
Index Symmetry in Multiple Dimensions
A B C
D E F
G H I
A B C
D E F
G H I
Consistent
Consistent
A B C
D E F
G H I
A B C
D E F
G H I
Inconsistent
Inconsistent
24
Incompleteness of Double Lex
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Swap 2 columns Swap row 1 and 3
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25
Enforcing Lexicographic Ordering

• We have developed a linear time algorithm for
  enforcing generalized arc-consistency on a
  lexicographic ordering constraint between two
  vectors of variables.
• Experiments show that in some cases it is vastly
  superior to previous consistency algorithms, both
  in time and in amount pruned.

26
Part III Constraint Spreadsheets

• Matrix of decision variables can be viewed and
  manipulated through a spreadsheet-style
  interface.
• Simple methods for imposing constraints on
  multiple rows/columns w/out iteration or
  recursion.
• Facilitate mixed-initiative solving and what-if
  reasoning.
• Endow spreadsheets with solving in addition to
  calculation.
• Object-oriented hierarchy of style sheets for
  solving common problem types.

27
Conclusion

• Constraint technology has made impressive
  progress, yet available toolkits support
  constraint programming at only a low level of
  abstraction.
• First steps to bring technology to the masses
• Develop high-level language supported with
  automatic refinement and reformulation,
  especially to matrix models.
• Efficient methods for solving matrix models,
  especially for breaking index symmetry.
• Constraint spreadsheets.