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Lecture3: Search 2

Constraint Satisfaction Problems

Definition

A constraint satisfaction problem \(P = (X,D,C)\)

  • Variables: \(X = \{X_1,\cdots,X_n\}\)
  • Domains: \(D = \{D_1,\cdots, D_n\}\)
  • Constraints: \(C\)

An assignment of values to some or all of the variables

\[ \{X_i = v_i, X_j = v_j, \cdots\} \]

A complete assignmet is one in which every variable is assigned

A solution to a CSP is a consistent and complete assignment.

Example

  • Map Coloring
  • N-Queens
  • Sudoku
  • The Waltz Algorithm
    • Interpreting line drawings of solid polyhedra

Constraint Graphs

  • Binary CSP: each constraint relates at most two variables

  • Binary constraint graph: nodes are variables, arcs show constraints

  • General-purpose CSP Algorithms

    • Use the graph structure to speed up search

Real-World CSPs

  • Many real-world problems involve real-valued variables

Standard Search Formulation

Standard search formulation of CSPs: - States: the values assigned so far (partial assignments) - Initial state: the empty assignment - Successor function:assign a value to an unassigned variable - Goal test:complete and satisfy all constraints

Backtracking Search

Idea 1: One variable at a time

  • Variable assignments are commutative, so fix ordering

Idea: 2: Check constraints as you go

Backtracking Search = DFS + variable-ording + fail-on-violation

Dynamic Ordering

Choosing an Unassigned Variable

Key idea: most constrained variable

Order Values of a Selected Variable

Key idea: least constrained value

Conflicting Heuristics?

  • Most constrained variable (MCV)
  • Least constrained value (LCV)

Combining these ordering ideas makes 1000 queens feasible

Arc Consistency

Interleaving Search andInference

  • Filtering: Keep track of domains for unassigned variables and infer new domain reductions

"Inference": check on the graph

Filtering: Forward Checking

"one-setp lookahead"

Arc Consistency

Definition

An arc \(X_i \to X_j\) is consistent iff for every \(x_i \in \mathrm{Domains_i}\), there exist \(x_j \in \mathrm{Domains_j}\) which can be assigned without violating a constraint

  • Enforce arc consistency: Remove values from \(Domains_i\) to make arc conststent
  • Forward checking: Enforcing consistency of arcs pointing to each new assignment.

AC-3

  • repeatedly enforce arc consistency on constraint chains

  • Assume a CSP with

    • \(n\) variables, each with domain size at most \(d\), \(c\) binary constraints
    • The complexity of AC-3: \(O(c\cdot d \cdot d^2)\)

Problem Structure

Tree Structured CSPs

Theorem

if the constraint graph has no loops, the CSP can be solved in linear time $O(nd^2)

  • Compare to general CSPs, where worst-case time is \(O(d^n)\)

Hill Climbing

  • Sometimes called greedy local search

  • Simple and general idea

    • start wherever
    • Repeat: move to the best neighboring state
    • If no neighbors better than current, quit

  • Problems:

    • Local maximum, shoulder: "flat local maximum"

Simulated Annealing

Basic idea: greedily keep \(k\) states at all times

  • Information is shared among \(k\) search threads.

Genetic Algorithm