Chapter 5 Constraint Satisfaction Problems

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Chapter 5 Constraint Satisfaction Problems
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Outline

• Constraint Satisfaction Problems (CSPs)
  representation
• Example map-coloring
• Backtracking search for CSPs
• Problem structure and problem decomposition
• Local search for CSPs

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CSP Representation

• Standard search problems
• state is a black box any old data structure
  that supports heuristic function, evaluation
  function, and goal test.
• Constraint Satisfaction Problems
• a set of variables and a set
  of constraints
• Each Xi has a domain Di of possible values
• Each Ci specifies allowable combinations of
  values for subsets of variables
• state is defined by an assignment of values to
  some or all of the variables
• consistent an assignment that does not violate
  any constraints
• complete assignment every variable is mentioned
• solution a complete assignment that satisfies
  all the constraints

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Example map-coloring
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Example map-coloring (contd)
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Constraint graph

• Binary CSP each constraint relates at most two
  variables
• Constraint graph nodes are variables, arcs show
  constraints

• General-purpose CSP algorithms use graph
  structure to speed up search.
• e.g., Tasmania is an independent subproblem.

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Varieties of CSPs
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Varieties of CSPs (contd)
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Example cryptarithmetic
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Real world CSPs

• Assignment problems
• e.g., who teaches what class?
• Timetabling problems
• e.g., which class is offered when and where?
• Transportation scheduling
• Hardware configuration
• Floorplanning
• Many real-world problems involve real-valued
  variables

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Standard search formulation (incremental)
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Backtracking search
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Backtracking example
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Improving backtracking efficiency

• General-purpose methods can give huge gains in
  speed
• Which variable should be assigned next?
• In what order should its values be tried?
• Can we detect inevitable failure early?
• Can we take advantage of problem structure?

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Minimum remaining values (MRV)

• MRV
• choose the variable with the fewest legal values
• also called most constrained variable

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Least constraining value

• Given a variable, choose the lest constraining
  value
• the one the rules out the fewest values in the
  remaining variables
• leave the maximum flexibility for subsequent
  variable assignments

• Combining these heuristics makes 1000 queens
  feasible

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Forward checking

• Keep track of remaining legal values for
  unassigned variables
• deletes from its domain any value inconsistent
• terminate search when any variable has no legal
  values

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Constraint propagation

• Forward checking propagates information from
  assigned variables, but does not provide early
  detection for all failures

• NT and SA cannot both be blue!
• Constraint propagation repeatedly enforces
  constraints locally

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Arc consistency

• Simplest form of propagation makes each arc
  consistent

directed arc
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Problem structure

• Tasmania and mainland are independent subproblems
• Identifiable as connected components of
  constraint graph

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Tree-structured CSPs
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Algorithm for tree-structured CSPs
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Nearly tree-structured CSPs
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Nearly tree-structured CSPs Tree Decomposition

• Every variable in the original problem appears in
  at least one of the subproblems.
• If two variables are connected by a constraint in
  the original problem, they must appear together
  (along with the constraint) in at least one of
  the subproblems
• If a variable appears in two subproblems in the
  tree, it must appear in every subproblem along
  the path connecting those subproblems. (i.e.,
  must have the same value)

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Iterative algorithms for CSPs

• Hill-climbing, simulated annealing typically work
  with complete states, i.e., all variables
  assigned
• To apply to CSPs
• allow states with unsatisfied constraints
• operators reassign variable values
• Variable selection randomly select any
  conflicted variable
• Value selection by min-conflicts heuristic
• choose value that violates the fewest constraints
• e.g., hill-climbing with h(n) total number of
  violated constraints

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Example 4-Queens
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Performance of min-conflicts
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Summary

• CSPs are a special kind of problem
• states defined by values of a fixed set of
  variables
• goal test defined by constraints on variable
  values
• Backtracking depth-first search with one
  variable assigned per node
• Variable ordering and value selection heuristics
  helps significantly
• Forward checking prevents assignments that
  guarantee later failure
• Constraint propagation (e.g., arc consistency)
  does additional work to constraint values and
  detect inconsistencies
• The CSP representation allows analysis of problem
  structure
• Tree-structured CSPs can be solved in linear time
• Iterative min-conflicts is usually effective in
  practice