Constraint Satisfaction

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• I am Patrick Prosser
• I am a senior lecturer
• at Glasgow (and Strathclyde till mid April)
• I teach algorithms data structures in java
• I am a member of
• the algorithms group
• the apes (distributed, not disbanded)
• I am a Glaswegian

This is all that I am allowed to tell you
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Constraint Satisfaction
Patrick Prosser

• What is it?
• Who cares (any applications?)
• Whats the challenge?

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Constraint Satisfaction

• 15 years research in 15 minutes

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An example
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Make a crossword puzzle! Given the above grid and
a dictionary, fill it. Then go get the clues (not
my problem)
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An example
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1A
4D
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4A
2D
1A 1 across 4D 4 down 2D 2
down 4A 4 across 7D 7 down
7D
Variables
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An example
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1A
4D
4
4A
2D
1A-4D 4th of 1A equals 1st of 4D 1A-2D 2nd of
1A equals 1st of 2D 2D-4A 4th of 2D equals 2nd
of4D 4D-4A 4th of 4A equals 4th of 4D 4A-7D 7th
of 4A equals 2nd of 7D
7D
Constraints
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An example
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1A
4D
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4A
2D
1A any 6 letter word 4A any 8 letter word 4D
any 5 letter word 2D any 7 letter word 7D any 3
letter word
7D
Domains (also unary constraints!)
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An example
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1A
4D
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4A
2D
Find an assignment of values to variables, from
their domains, such that the constraints are
satisfied (or show that no assignment exists)
7D
A CSP!
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An example
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1A
4D
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4A
2D
Choose a variable Assign it a value Check
compatibility If not compatible try a new
value If no values remain re-assign previous
variable
7D
Good old fashioned BT!
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Questions?
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1A
4D
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4A
2D
What variable should I choose?
7D
What value should I choose?
What reasoning can I do when making an assignment?
What reasoning can I do on a dead end?
Decisions, decisions!
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Search Algorithms

• BT (1960s)
• BM (1979)
• FC (1980)
• MAC (1979)
• CBJ (1993)
• DB (1993)
• Combinations of the above (1993-),
• LDS (1995)
• DDS (1997)
• Incomplete Search (min conflicts?)

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Heuristics

• Fail First (aka minimum remaining values)
• are you serious? Do you really want to fail?

• Consider topology of constraint graph
• degree of vertex, bandwidth of ordering,

• Regret?

• Minimise constrainedness?

• Static or dynamic?

• we aint even considered values!!

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What makes a problem hard?

• Size?
• More words in our crossword?
• Bigger dictionary?
• Tightness?
• A smaller dictionary?
• More crossings between words
• Phase transition phenomena

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Constraint Propagation

• Pre-process, remove inconsistencies
• for each value, is there a supporting value?
• Can I specialise this test/filter for the
  constraint
• think of all-diff and edgeFinding

• What level of consistency is worthwhile?
• Arc/path/inverse/singleton

• Should we embed these in the search process?
• MAC -- AC in BT, the heart of CP

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Inside Search

• As search proceeds the remaining subproblem gets
  smaller

• Does the subproblem get easier? Harder?
• What features make it easier or harder
• Can we design algorithms to exploit this?

• Are the heuristic genuinely dynamic?

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Representation

• If we change the representation of the problem
  what effect will this have?
• Could it make constraint propagation more
  effective?
• Could it give more information to the
  heuristics?
• Will it make search easier?

• Three examples
• change arity of constraints
• view the problem as something else
• jssp at vrp
• interchange variables and values (think 3COL)

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Applications

• Constraint Programming Languages
• CHIP, ILOG Solver (G), Claire, eClipSe, ...

• Scheduling (ILOG Schedule, Eclaire)
• Transportation (ILOG Dispatcher, BT WFM)
• Frequency Allocation
• Gate Management
• Car Sequencing
• Design (car interiors)
• Configuration (discs, computers, )
• Timetabling
• Real-time resource allocation (weapon systems)
• Folding

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Challenges

• Combine CP with OR

• Push the existing technology to larger problems

• Design new constraints/propagators

• Understand the role of representation

• Better understanding of search process

• Better understanding of pre-processing

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