Lecture 6: Constraint Satisfaction Problems

Lecture 6: Constraint Satisfaction Problems¶

AIMA Chapter 6 — 1 hour¶

Why CSPs Matter¶

• Scheduling, assignment, design, configuration

• Same abstraction: variables, domains, constraints

• Inference (arc consistency) + search (backtracking)

• Local search (min-conflicts) often wins in practice

Learning Objectives¶

• Define CSPs: variables, domains, constraints

• Apply constraint propagation: arc consistency, path consistency

• Implement backtracking search with heuristics

• Use local search for CSPs

• Exploit problem structure

CSP Formulation¶

• Variables: X₁, X₂, ..., Xₙ

• Domains: D₁, D₂, ..., Dₙ (possible values)

• Constraints: Restrictions on variable combinations

• Solution: Assignment satisfying all constraints

Example: Map Coloring¶

• Variables: Regions (e.g., WA, NT, SA, Q, NSW, V, T)

• Domain: {red, green, blue}

• Constraints: Adjacent regions different colors

• Solution: Valid coloring

Example: N-Queens¶

• Variables: Queen position (row or column) for each queen

• Domain: {1, 2, ..., N}

• Constraints: No two attack each other

• Binary constraints: Pairwise

Constraint Types¶

• Unary: Single variable (e.g., SA ≠ red)

• Binary: Two variables (e.g., WA ≠ NT)

• Higher-order: 3+ variables

• Global: All-different, etc.

Constraint Propagation¶

• Inference: Reduce domains before search

• Node consistency: Unary constraints

• Arc consistency: For each (X,Y), for each x in D(X), ∃ y in D(Y) satisfying (X,Y)

AC-3 Algorithm¶

• Maintain queue of arcs to check

• When (X,Y) reduces D(X), add arcs (Z,X) for all Z

• Repeat until no changes

• O(n²d³) for n variables, domain size d

Backtracking Search¶

• Depth-first: Assign variables one by one

• Backtrack: When constraint violated, try next value

• Improvement: Use inference when assigning

Variable Ordering¶

• MRV (Minimum Remaining Values): Choose variable with fewest choices

• Degree heuristic: Choose variable involved in most constraints

• Fail-first: Reduces branching factor

Value Ordering¶

• Least Constraining Value: Choose value that rules out fewest values for neighbors

• Enables more solutions downstream

Interleaving Search and Inference¶

• Forward checking: After assigning X, remove inconsistent values from neighbors

• MAC (Maintaining Arc Consistency): Full AC-3 after each assignment

• MAC: Stronger pruning, more inference cost

Intelligent Backtracking¶

• Conflict-directed backtracking: Jump to variable that caused failure

• Backjumping: Don’t try sibling values of same variable

• Constraint learning: Add nogood constraints

Local Search for CSPs¶

• State: Complete assignment (may violate constraints)

• Min-conflicts: Pick variable in conflict, assign value that minimizes conflicts

• Very effective for many CSPs (e.g., N-queens, scheduling)

Problem Structure¶

• Tree-structured CSP: No cycles → O(nd²) with arc consistency

• Cutset conditioning: Instantiate cutset, solve tree

• Tree decomposition: Decompose into tree of subproblems

Summary¶

• CSP: Variables, domains, constraints

• Propagation: Arc consistency (AC-3)

• Backtracking: MRV, LCV, forward checking, MAC

• Local search: Min-conflicts

• Structure: Trees, cutsets, decomposition

References¶

• Russell & Norvig, AIMA 4e, Ch. 6

• Chapter PDF: chapters/chapter-06.pdf

• aima-python: csp.ipynb

Questions?¶

Next lecture: Logical Agents (Chapter 7)