CS 188: Artificial Intelligence
Spring 2006
Lecture 4: CSPs
9/7/2006
Dan Klein – UC Berkeley
Many slides over the course adapted from either Stuart
Russell or Andrew Moore
Announcements
ƒ Reminder:
ƒ Project 1.1 is due Friday at 11:59pm!
ƒ Check web page for this week’s office hours
ƒ Sections this Monday
ƒ Can go to any of them, or multiple (unless over
capacity of room)
ƒ Dan / John back late today
ƒ Don’t forget about the newsgroup
ƒ Good for course questions
ƒ Good for finding partners
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Constraint Satisfaction Problems
ƒ Standard search problems:
ƒ State is a “black box”: any old data structure
ƒ Goal test: any function over states
ƒ Successors: any map from states to sets of states
ƒ Constraint satisfaction problems (CSPs):
ƒ State is defined by variables Xi with values from a
domain D (sometimes D depends on i)
ƒ Goal test is a set of constraints specifying
allowable combinations of values for subsets of
variables
ƒ Simple example of a formal representation
language
ƒ Allows useful general-purpose algorithms with
more power than standard search algorithms
Example: N-Queens
ƒ Formulation 1:
ƒ Variables:
ƒ Domains:
ƒ Constraints
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Example: N-Queens
ƒ Formulation 2:
ƒ Variables:
ƒ Domains:
ƒ Constraints:
… there’s an even better way! What is it?
Example: Map-Coloring
ƒ Variables:
ƒ Domain:
ƒ Constraints: adjacent regions must have
different colors
ƒ Solutions are assignments satisfying all
constraints, e.g.:
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Example: The Waltz Algorithm
ƒ The Waltz algorithm is for interpreting line drawings of
solid polyhedra
ƒ An early example of a computation posed as a CSP
?
ƒ Look at all intersections
ƒ Adjacent intersections impose constraints on each other
Waltz on Simple Scenes
ƒ Assume all objects:
ƒ Have no shadows or cracks
ƒ Three-faced vertices
ƒ “General position”: no junctions
change with small movements of
the eye.
ƒ Then each line on image is
one of the following:
ƒ Boundary line (edge of an
object) (→) with right hand of
arrow denoting “solid” and left
hand denoting “space”
ƒ Interior convex edge (+)
ƒ Interior concave edge (-)
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Legal Junctions
ƒ Only certain junctions are
physically possible
ƒ How can we formulate a CSP to
label an image?
ƒ Variables: vertices
ƒ Domains: junction labels
ƒ Constraints: both ends of a line
should have the same label
x
(x,y) in
y
,
,…
Constraint Graphs
ƒ Binary CSP: each constraint
relates (at most) two variables
ƒ Constraint graph: nodes are
variables, arcs show
constraints
ƒ General-purpose CSP
algorithms use the graph
structure to speed up search.
E.g., Tasmania is an
independent subproblem!
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Example: Cryptarithmetic
ƒ Variables:
ƒ Domains:
ƒ Constraints:
Varieties of CSPs
ƒ Discrete Variables
ƒ Finite domains
ƒ Size d means O(dn) complete assignments
ƒ E.g., Boolean CSPs, including Boolean satisfiability (NP-complete)
ƒ Infinite domains (integers, strings, etc.)
ƒ E.g., job scheduling, variables are start/end times for each job
ƒ Need a constraint language, e.g., StartJob1 + 5 < StartJob3
ƒ Linear constraints solvable, nonlinear undecidable
ƒ Continuous variables
ƒ E.g., start/end times for Hubble Telescope observations
ƒ Linear constraints solvable in polynomial time by LP methods
(see cs170 for a bit of this theory)
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Varieties of Constraints
ƒ Varieties of Constraints
ƒ Unary constraints involve a single variable (equiv. to shrinking domains):
ƒ Binary constraints involve pairs of variables:
ƒ Higher-order constraints involve 3 or more variables:
e.g., cryptarithmetic column constraints
ƒ Preferences (soft constraints):
ƒ
ƒ
ƒ
ƒ
E.g., red is better than green
Often representable by a cost for each variable assignment
Gives constrained optimization problems
(We’ll ignore these until we get to Bayes’ nets)
Real-World CSPs
ƒ Assignment problems: e.g., who teaches what class
ƒ Timetabling problems: e.g., which class is offered when
and where?
ƒ Hardware configuration
ƒ Spreadsheets
ƒ Transportation scheduling
ƒ Factory scheduling
ƒ Floorplanning
ƒ Many real-world problems involve real-valued
variables…
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Standard Search Formulation
ƒ Standard search formulation of CSPs (incremental)
ƒ Let's start with the straightforward, dumb approach, then
fix it
ƒ States are defined by the values assigned so far
ƒ Initial state: the empty assignment, {}
ƒ Successor function: assign a value to an unassigned variable
ƒ Goal test: the current assignment is complete and satisfies all
constraints
Search Methods
ƒ What does BFS do?
ƒ What does DFS do?
ƒ [ANIMATION]
ƒ What’s the obvious problem here?
ƒ What’s the slightly-less-obvious problem?
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Backtracking Search
ƒ Idea 1: Only consider a single variable at each point:
ƒ
ƒ
ƒ
ƒ
Variable assignments are commutative
I.e., [WA = red then NT = green] same as [NT = green then WA = red]
Only need to consider assignments to a single variable at each step
How many leaves are there?
ƒ Idea 2: Only allow legal assignments at each point
ƒ I.e. consider only values which do not conflict previous assignments
ƒ Might have to do some computation to figure out whether a value is ok
ƒ Depth-first search for CSPs with these two improvements is called
backtracking search
ƒ [ANIMATION]
ƒ Backtracking search is the basic uninformed algorithm for CSPs
ƒ Can solve n-queens for n ≈ 25
Backtracking Search
ƒ What are the choice points?
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Backtracking Example
Improving Backtracking
ƒ General-purpose ideas can give huge gains in
speed:
ƒ
ƒ
ƒ
ƒ
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
ƒ Minimum remaining values (MRV):
ƒ Choose the variable with the fewest legal values
ƒ Why min rather than max?
ƒ Called most constrained variable
ƒ “Fail-fast” ordering
Degree Heuristic
ƒ Tie-breaker among MRV variables
ƒ Degree heuristic:
ƒ Choose the variable with the most constraints on
remaining variables
ƒ Why most rather than fewest constraints?
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Least Constraining Value
ƒ Given a choice of variable:
ƒ Choose the least constraining
value
ƒ The one that rules out the fewest
values in the remaining variables
ƒ Note that it may take some
computation to determine this!
ƒ Why least rather than most?
ƒ Combining these heuristics
makes 1000 queens feasible
Forward Checking
NT
WA
SA
Q
NSW
V
ƒ Idea: Keep track of remaining legal values for
unassigned variables
ƒ Idea: Terminate when any variable has no legal values
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Constraint Propagation
NT
WA
SA
Q
NSW
V
ƒ Forward checking propagates information from assigned to
unassigned variables, but doesn't provide early detection for all
failures:
ƒ NT and SA cannot both be blue!
ƒ Why didn’t we detect this yet?
ƒ Constraint propagation repeatedly enforces constraints (locally)
Arc Consistency
NT
WA
SA
Q
NSW
V
ƒ Simplest form of propagation makes each arc consistent
ƒ X → Y is consistent iff for every value x there is some allowed y
ƒ
ƒ
ƒ
ƒ
If X loses a value, neighbors of X need to be rechecked!
Arc consistency detects failure earlier than forward checking
What’s the downside of arc consistency?
Can be run as a preprocessor or after each assignment
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Arc Consistency
ƒ Runtime: O(n2d3), can be reduced to O(n2d2)
ƒ … but detecting all possible future problems is NP-hard – why?
Problem Structure
ƒ Tasmania and mainland are
independent subproblems
ƒ Identifiable as connected
components of constraint graph
ƒ Suppose each subproblem has c
variables out of n total
ƒ Worst-case solution cost is
O((n/c)(dc)), linear in n
ƒ E.g., n = 80, d = 2, c =20
ƒ 280 = 4 billion years at 10 million
nodes/sec
ƒ (4)(220) = 0.4 seconds at 10 million
nodes/sec
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Tree-Structured CSPs
ƒ Theorem: if the constraint graph has no loops, the CSP can be
solved in O(n d2) time (next slide)
ƒ Compare to general CSPs, where worst-case time is O(dn)
ƒ This property also applies to logical and probabilistic reasoning: an
important example of the relation between syntactic restrictions and
the complexity of reasoning.
Tree-Structured CSPs
ƒ Choose a variable as root, order
variables from root to leaves such
that every node's parent precedes
it in the ordering
ƒ For i = n : 2, apply RemoveInconsistent(Parent(Xi),Xi)
ƒ For i = 1 : n, assign Xi consistently with Parent(Xi)
ƒ Runtime: O(n d2)
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Nearly Tree-Structured CSPs
ƒ Conditioning: instantiate a variable, prune its neighbors' domains
ƒ Cutset conditioning: instantiate (in all ways) a set of variables such
that the remaining constraint graph is a tree
ƒ Cutset size c gives runtime O( (dc) (n-c) d2 ), very fast for small c
Iterative Algorithms for CSPs
ƒ Greedy and local methods typically work with “complete”
states, i.e., all variables assigned
ƒ To apply to CSPs:
ƒ Allow states with unsatisfied constraints
ƒ Operators reassign variable values
ƒ Variable selection: randomly select any conflicted
variable
ƒ Value selection by min-conflicts heuristic:
ƒ Choose value that violates the fewest constraints
ƒ I.e., hill climb with h(n) = total number of violated constraints
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Example: 4-Queens
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ƒ
ƒ
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States: 4 queens in 4 columns (44 = 256 states)
Operators: move queen in column
Goal test: no attacks
Evaluation: h(n) = number of attacks
Performance of Min-Conflicts
ƒ Given random initial state, can solve n-queens in almost constant
time for arbitrary n with high probability (e.g., n = 10,000,000)
ƒ The same appears to be true for any randomly-generated CSP
except in a narrow range of the ratio
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Summary
ƒ
CSPs are a special kind of search problem:
ƒ
ƒ
States defined by values of a fixed set of variables
Goal test defined by constraints on variable values
ƒ
Backtracking = depth-first search with one legal variable assigned per node
ƒ
Variable ordering and value selection heuristics help significantly
ƒ
Forward checking prevents assignments that guarantee later failure
ƒ
Constraint propagation (e.g., arc consistency) does additional work to constrain
values and detect inconsistencies
ƒ
The constraint graph representation allows analysis of problem structure
ƒ
Tree-structured CSPs can be solved in linear time
ƒ
Iterative min-conflicts is usually effective in practice
Local Search Methods
ƒ Queue-based algorithms keep fallback
options (backtracking)
ƒ Local search: improve what you have until
you can’t make it better
ƒ Generally much more efficient (but
incomplete)
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Types of Problems
ƒ Planning problems:
ƒ We want a path to a solution
(examples?)
ƒ Usually want an optimal path
ƒ Incremental formulations
ƒ Identification problems:
ƒ We actually just want to know what
the goal is (examples?)
ƒ Usually want an optimal goal
ƒ Complete-state formulations
ƒ Iterative improvement algorithms
Hill Climbing
ƒ Simple, general idea:
ƒ Start wherever
ƒ Always choose the best neighbor
ƒ If no neighbors have better scores than
current, quit
ƒ Why can this be a terrible idea?
ƒ Complete?
ƒ Optimal?
ƒ What’s good about it?
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Hill Climbing Diagram
ƒ Random restarts?
ƒ Random sideways steps?
Simulated Annealing
ƒ Idea: Escape local maxima by allowing downhill moves
ƒ But make them rarer as time goes on
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Simulated Annealing
ƒ Theoretical guarantee:
ƒ Stationary distribution:
ƒ If T decreased slowly enough,
will converge to optimal state!
ƒ Is this an interesting guarantee?
ƒ Sounds like magic, but reality is reality:
ƒ The more downhill steps you need to escape, the less
likely you are to every make them all in a row
ƒ People think hard about ridge operators which let you
jump around the space in better ways
Beam Search
ƒ Like greedy search, but keep K states at all
times:
Greedy Search
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ƒ
ƒ
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Beam Search
Variables: beam size, encourage diversity?
The best choice in MANY practical settings
Complete? Optimal?
Why do we still need optimal methods?
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Genetic Algorithms
ƒ Genetic algorithms use a natural selection metaphor
ƒ Like beam search (selection), but also have pairwise
crossover operators, with optional mutation
ƒ Probably the most misunderstood, misapplied (and even
maligned) technique around!
Example: N-Queens
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ƒ
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Why does crossover make sense here?
When wouldn’t it make sense?
What would mutation be?
What would a good fitness function be?
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Continuous Problems
ƒ Placing airports in Romania
ƒ States: (x1,y1,x2,y2,x3,y3)
ƒ Cost: sum of squared distances to closest city
Gradient Methods
ƒ How to deal with continous (therefore infinite)
state spaces?
ƒ Discretization: bucket ranges of values
ƒ E.g. force integral coordinates
ƒ Continuous optimization
ƒ E.g. gradient ascent
Image from vias.org
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