i may require adding new constraints, except for

1
When A Little Reasoning Saves A Lot of Hard Work
Robert Woodward Berthe Y. Choueiry Constraint
Systems Laboratory Department of Computer
Science Engineering University of
Nebraska-Lincoln
2. Techniques Contributions
1. Context Focus
3. Illustration Minesweeper as a CSP

• Relevance
• Motivates research
• Facilitates teaching of complex concepts
  mechanisms
• Helps in outreach recruiting
• Demystifies human fascination with puzzles

• Algorithms for Constraint Propagation enforce
  relational consistency properties R(i,m)C where
• m is the number of constraints considered
• i is the number of variables considered

• A Constraint Satisfaction Problem (CSP) is
  defined by

• A set of decisions to make (variables)
• A set of choices for each variable (values,
  domain)
• A set of constraints restricting the allowable
  combinations of values (tuples) to variables

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A
B
C
D
E
F
G
0,1
0,1
0,1
0,1
0,1
0,1
0,1
Modeling Minesweeper with Constraints

• i may require adding new constraints, except
  for
• i 1 ? domain filtering
• i ? ? constraint filtering

Exactly 2 mines
A B C D E J I
0 0 0 0 0 1 1
0 0 0 0 1 0 1

1 1 0 0 0 0 0
A B C D
0 0 0 1
1 0 1 1
0 1 1 0
1 1 1 0
B C E F
0 0 1 1
0 1 0 0
1 1 1 0
0 0 0 1
A D E F G
0 1 1 1 1
0 0 1 0 1
1 1 0 1 1
1 0 1 1 1
The task is find one solution (i.e., an
assignment of values to variables satisfying all
constraints) or all solutions
Exactly 3 mines

• Applications include scheduling resource
  allocation, design product configuration,
  software hardware verification, Puzzles, etc.

R(1,1)C
R(1,3)C
R(?,2)C
R(?,3)C
J I H G F E D
0 0 0 0 1 1 1
0 0 0 1 1 0 1

1 1 1 0 0 0 0
A
E
G
0,1
0,1
0,1

• CSPs are solved using
• Search it laboriously enumerates combinations of
  assignments of values to variables. Search can
  be done in a smart way, but is in general
  tedious (i.e., exponential cost).
• Constraint Propagation it thinks about the
  constraints to remove values (from variables)
  tuples (from constraints) that cannot
  participate in any solution.

A B C D
0 0 0 1
1 0 1 1
0 1 1 0
1 1 1 0
B C E F
0 0 1 1
0 1 0 0
1 1 1 0
0 0 0 1
A D E F G
0 1 1 1 1
0 0 1 0 1
1 1 0 1 1
1 0 1 1 1
A B C D
0 0 0 1
1 0 1 1
0 1 1 0
1 1 1 0
B C E F
0 0 1 1
0 1 0 0
1 1 1 0
0 0 0 1
A D E F G
0 1 1 1 1
0 0 1 0 1
1 1 0 1 1
1 0 1 1 1
Domain Filtering R(1,m)C Domain Filtering R(1,m)C
Polynomial space only for m2 Otherwise, exponential space
Two linear-space algorithms vvpSearch suitable for loose constraints AllSearch suitable for tight constraints
Constraint Filtering R(?,m)C Constraint Filtering R(?,m)C
Dual-AC3 only for m2 Otherwise, none existed
One exponential three linear-space algorithms Join-R(, m)C exponential space, conceptual Dual-AC2009 only for m2 Search-R(?,m)C suitable for loose constraints AllSearch-R(?,m)C suitable for tight constraints
A C
R R
R G
G R
G G

• Constraint propagation operates locally. It is
  cheap (i.e., polynomial time) can
  considerably reduce search effort.
• Thus, a little thinking can save a lot of hard
  work

• The focus of our research is the development of
  new algorithms for constraint propagation

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Acknowledgments

• Robert Woodward
• Was supported by UCARE during 20072008
  20082009.
• Is the recipient of a Barry M. Goldwater
  Scholarship for 20082010.
• Work on Minesweeper as a CSP was started by Josh
  Snyder continued by Ken Bayer under CAREER
  Award 0133568 from the National Science
  Foundation.
• Ongoing evaluations are in collaboration with
  Shant Karakashian.