Lecture 10 CSP II Improving Backtracking

Recap: Constraint Satisfaction Problems §

• Standard Search Problems:
  • State is a “black box” (arbitrary data structure)
  • Goal test can be any function over states
• Constraint Satisfaction Problems (CSPs):
  • A special subset of search problems
  • State is defined by variables $X_i$ with values from a domain $D$
  • Goal Test is a set of constraints specifying allowable combinations of values for subsets of variables.

Recap: Backtracking Search §

• Basic uninformed algorithm for solving CSPs
• Idea 1: One variable at a time
  • Variable assignments commutative
  • Fixed ordering of variables and only considering assignment to a single variable at each step
• Idea 2: Check constraints as you go
  • “Incremental Goal Test”

Recap - Filtering: Forward Checking §

• Filtering: Keep track of domains for unassigned variables and cross off bad options.
• Forward Checking: Cross off values that violate a constraint when a value assignment is added to the current state.

Recap: Consistency of a Single Arc §

• Arc $X\to Y$ is consistent iff for every $x$ in the tail, there exists some $y$ in the head which could be assigned without violating a constraint.
  • $\forall x\exists yConsistent(x,y)$
• Forward Checking: Enforcing consistency of arcs from unassigned variables (i.e. X) pointing to the variable with a new assignment (i.e. Y)
• Important: If $X$ loses a value, neighbors of $X$ need to be rechecked!
• AC-3 Algorithm

CSP Ordering §

Ordering of Variables §

Minimum Remaining Values
Degree Heuristic
Least Constraining Value

Main Idea:

• Which variable should be assigned next?
  • MRV, Degree
• In what order should it’s values be tried?
  • LCV

Minimum Remaining Values §

• Variable Ordering: The Minimum Remaining Values (MRV) heuristic
  • Choose variable with the fewest number of remaining values in it’s domain
  • Also called “most constrained variable”
  • “Fail-Fast” ordering

Degree Heuristic §

• Degree Heuristic: Select variable involved in the largest number of constraints.
  • Makes the largest number of other variables more “constrained” and enables early failures indirectly

Least Constraining Value (LCV) §

• Value Ordering: The least constraining value heuristic
  • Given a choice of variable, choose the least constraining value.
  • Value that rules out fewest values in other unassigned variables.
• Why Least?
  • Once variable to be assigned is determined, we only need one consistent solution.

CSP Problem Structure §

• Extreme Case: Independent Subproblems
• Independent subproblems are identifiable as ’connected components’ of constraint graph
• Decomposing graph into subproblems, then can save in time complexity.

Tree-Structured CSPs §

• Theorem: If constraint graph has no loops (tree structure), then can be solved in $O(n{d}^2)$
  • Compare to general CSPs, where worst case is $O(d^n)$
• Algorithm:
  • Order: Choose root variable, order variables so that parents precede children.
  • Enforce arc consistency from leaf to root
    • Parent($X_i$) $\to X_i$
    • Runtime: $O(n{d}^2)$

Nearly Tree-Structured CSPs §

• Cutset: set of variables such that after removing variables in this set and constraints associated, then the constraint graph becomes a tree.
• Cutset Conditioning: Instantiate the variables in the cutset, update the domains for the remaining variables, and solve CSP subproblem by remaining constraint tree.
• When number of variables in cutset is $c$, the worst-case runtime is:
  • $O((d^c)(n-c)d^2)$, fast for small c.